Solve Linear Inequalities

Graph Inequalities on the Number Line

What number would make the inequality \(x>3\) true? Are you thinking, “x could be four”? That’s correct, but x could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality \(x>3.\)

We show all the solutions to the inequality \(x>3\) on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.

We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express \(x>3\) as \((3,\infty ).\) The symbol \(\infty\) is read as “infinity.” It is not an actual number.

shows both the number line and the interval notation.

We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.

The inequality \(x\le 1\) means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at \(x=1.\) We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See .

There is no lower end to those numbers. We write \(x\le 1\) in interval notation as \((\text{-}\infty ,1].\) The symbol \(\text{-}\infty\) is read as “negative infinity.” shows both the number line and interval notation.

Example

Try it.

Graph each inequality on the number line and write in interval notation.

ⓐ \(x\ge -3\) ⓑ \(x<2.5\) ⓒ \(x\le -\frac{3}{5}\)

Solution

ⓐ

Shade to the right of \(-3,\) and put a bracket at \(-3.\)
Write in interval notation.

ⓑ

Shade to the left of 2.5 and put a parenthesis at 2.5.
Write in interval notation.

ⓒ

Shade to the left of \(-\frac{3}{5},\) and put a bracket at \(-\frac{3}{5}.\)
Write in interval notation.

Example

Try it.

Graph each inequality on the number line and write in interval notation.

ⓐ \(-3

Solution

ⓐ

Shade between \(-3\) and 4. Put a parentheses at \(-3\) and 4.
Write in interval notation.

ⓑ

Shade between \(-6\) and −1. Put a bracket at \(-6,\) and a parenthesis at −1.
Write in interval notation.

ⓒ

Shade between 0 and 2.5. Put a bracket at 0 and at 2.5.
Write in interval notation.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms, \(ax+bc,\) or \(ax+b\ge c.\)

When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:

Notice that the inequality sign stayed the same.

This leads us to the Addition and Subtraction Properties of Inequality.

What happens to an inequality when we divide or multiply both sides by a constant?

Let’s first multiply and divide both sides by a positive number.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. shows some common phrases that indicate inequalities.

\(>\)	\(\ge\)	\(<\)	\(\le\)
is greater than is more than is larger than exceeds	is greater than or equal to is at least is no less than is the minimum	is less than is smaller than has fewer than is lower than	is less than or equal to is at most is no more than is the maximum

Example

Try it.

Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

\[\text{Twenty-seven less than}\ x\ \text{is at least 48.}\]

Solution

Translate.
Solve—add 27 to both sides.
Simplify.
Graph on the number line.
Write in interval notation.

Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.

We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.

Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down.

Example

Try it.

Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the maximum number of tablets Dawn can buy
Step 3. Name what you are looking for.
Choose a variable to represent that quantity.	\(\text{Let}\ n=\text{the number of tablets.}\)
Step 4. Translate. Write a sentence that gives the information to find it.	$254.12 times the number of tablets is no more than $4,000.
Translate into an inequality.	\(254.12n\le 4000\)
Step 5. Solve the inequality. But n must be a whole number of tablets, so round to 15.	\(\begin{array}{l} \\ \\ n\le 15.74 \\ n\le 15\end{array}\)
Step 6. Check the answer in the problem and make sure it makes sense.
Rounding down the price to $250, 15 tablets would cost $3,750, while 16 tablets would be $4,000. So a maximum of 15 tablets at $254.12 seems reasonable.
Step 7. Answer the question with a complete sentence.	Dawn can buy a maximum of 15 tablets.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Inequalities, Number Lines, and Interval Notation
  \(x>a\ x\ge a\ x
• Linear Inequality
  • A linear inequality is an inequality in one variable that can be written in one of the following forms where a, b, and c are real numbers and \(a\ne 0:\)
    \[ax+bc,\ ax+b\ge c.\]
• Addition and Subtraction Property of Inequality
  • For any numbers a, b, and c, if \(a \[\begin{array}{llll}a+cb+c & & & a-c>b-c\end{array}\]
  • We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.
• Multiplication and Division Property of Inequality
  • For any numbers a, b, and c,
    \(\begin{array}{l}\text{multiply or divide by a}\ \text{positive} \\ \\ \\ \text{if}\ a0,\ \text{then}\ acb\ \text{and}\ c>0,\ \text{then}\ ac>bc\ \text{and}\ \frac{a}{c}>\frac{b}{c}. \\ \text{multiply or divide by a}\ \text{negative} \\ \\ \\ \text{if}\ abc\ \text{and}\ \frac{a}{c}>\frac{b}{c}. \\ \text{if}\ a>b\ \text{and}\ c<0,\ \text{then}\ ac
• Phrases that indicate inequalities
  

  \(>\)	\(\ge\)	\(<\)	\(\le\)
  is greater than is more than is larger than exceeds	is greater than or equal to is at least is no less than is the minimum	is less than is smaller than has fewer than is lower than	is less than or equal to is at most is no more than is the maximum

Solve Linear Inequalities

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line and write in interval notation.

Try it.

ⓐ \(x<-2\)
ⓑ \(x\ge -3.5\)
ⓒ \(x\le \frac{2}{3}\)

Try it.

ⓐ \(x>3\)
ⓑ \(x\le -0.5\)
ⓒ \(x\ge \frac{1}{3}\)

Solution

ⓐ

ⓑ
ⓒ

Try it.

ⓐ \(x\ge -4\)
ⓑ \(x<2.5\)
ⓒ \(x>-\frac{3}{2}\)

Try it.

ⓐ \(x\le 5\)
ⓑ \(x\ge -1.5\)
ⓒ \(x<-\frac{7}{3}\)

Solution

ⓐ

ⓑ

ⓒ

Try it.

ⓐ \(-5ⓑ \(-3\le x<1\)
ⓒ \(0\le x\le 1.5\)

Try it.

ⓐ \(-2ⓑ \(-5\le x<-3\)
ⓒ \(0\le x\le 3.5\)

Solution

ⓐ

ⓑ

ⓒ

Try it.

ⓐ \(-1ⓑ \(-3ⓒ \(-1.25\le x\le 0\)

Try it.

ⓐ \(-4ⓑ \(-5ⓒ \(-3.75\le x\le 0\)

Solution

ⓐ

ⓑ

ⓒ

Solve Linear Inequalities

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Try it.

ⓐ \(a+\frac{3}{4}\ge \frac{7}{10}\)
ⓑ \(8x>72\)
ⓒ \(20>\frac{2}{5}h\)

Try it.

ⓐ \(b+\frac{7}{8}\ge \frac{1}{6}\)
ⓑ \(6y<48\)
ⓒ \(40<\frac{5}{8}k\)

Solution

ⓐ

ⓑ

ⓒ

Try it.

ⓐ \(f-\frac{13}{20}<-\frac{5}{12}\)
ⓑ \(9t\ge -27\)
ⓒ \(\frac{7}{6}j\ge 42\)

Try it.

ⓐ \(g-\frac{11}{12}<-\frac{5}{18}\)
ⓑ \(7s<-28\)
ⓒ \(\frac{9}{4}g\le 36\)

Solution

ⓐ

ⓑ

ⓒ

Try it.

ⓐ \(-5u\ge 65\)
ⓑ \(\frac{a}{-3}\le 9\)

Try it.

ⓐ \(-8v\le 96\)
ⓑ \(\frac{b}{-10}\ge 30\)

Solution

ⓐ

ⓑ

Try it.

ⓐ \(-9c<126\)
ⓑ \(-25<\frac{p}{-5}\)

Try it.

ⓐ \(-7d>105\)
ⓑ \(-18>\frac{q}{-6}\)

Solution

ⓐ

ⓑ

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Try it.

\(4v\ge 9v-40\)

Try it.

\(5u\le 8u-21\)

Solution

Try it.

\(13q<7q-29\)

Try it.

\(9p>14p-18\)

Solution

Try it.

\(12x+3(x+7)>10x-24\)

Try it.

\(9y+5(y+3)<4y-35\)

Solution

Try it.

\(6h-4(h-1)\le 7h-11\)

Try it.

\(4k-(k-2)\ge 7k-26\)

Solution

Try it.

\(8m-2(14-m)\ge \text{}7(m-4)+3m\)

Try it.

\(6n-12(3-n)\le 9(n-4)+9n\)

Solution

Try it.

\(\frac{3}{4}b-\frac{1}{3}b<\frac{5}{12}b-\frac{1}{2}\)

Try it.

\(9u+5(2u-5)\ge 12(u-1)+7u\)

Solution

Try it.

\(\frac{2}{3}g-\frac{1}{2}(g-14)\le \frac{1}{6}(g+42)\)

Try it.

\(\frac{4}{5}h-\frac{2}{3}(h-9)\ge \frac{1}{15}(2h+90)\)

Solution

Try it.

\(\frac{5}{6}a-\frac{1}{4}a>\frac{7}{12}a+\frac{2}{3}\)

Try it.

\(12v+3(4v-1)\le 19(v-2)+5v\)

Solution

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Try it.

\(15k\le -40\)

Try it.

\(35k\ge -77\)

Solution

Try it.

\(23p-2(6-5p)>3(11p-4)\)

Try it.

\(18q-4(10-3q)<5(6q-8)\)

Solution

Try it.

\(-\frac{9}{4}x\ge -\frac{5}{12}\)

Try it.

\(-\frac{21}{8}y\le -\frac{15}{28}\)

Solution

Try it.

\(c+34<-99\)

Try it.

\(d+29>-61\)

Solution

Try it.

\(\frac{m}{18}\ge -4\)

Try it.

\(\frac{n}{13}\le -6\)

Solution

Translate to an Inequality and Solve

In the following exercises, translate and solve. Then graph the solution on the number line and write the solution in interval notation.

Try it.

Three more than h is no less than 25.

Try it.

Six more than k exceeds 25.

Solution

Try it.

Ten less than w is at least 39.

Try it.

Twelve less than x is no less than 21.

Solution

Try it.

Negative five times r is no more than 95.

Try it.

Negative two times s is lower than 56.

Solution

Try it.

Nineteen less than b is at most \(-22.\)

Try it.

Fifteen less than a is at least \(-7.\)

Solution

Solve Applications with Linear Inequalities

In the following exercises, solve.

Try it.

Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?

Try it.

The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?

Solution

A maximum of 14 people can safely ride in the elevator.

Try it.

Andre is looking at apartments with three of his friends. They want the monthly rent to be no more than $2,360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?

Try it.

Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $3.79. What is the maximum number of drinks she can buy with the gift card?

Solution

five drinks

Try it.

Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?

Try it.

Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least $1,500?

Solution

86 cars

Try it.

Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?

Try it.

Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her $3,475 per month plus 4% of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to pay all her expenses?

Solution

$16,875

Try it.

Andre has been offered an entry-level job. The company offered him $48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is $62,000. What would Andre’s total sales need to be for his pay to be at least as high as the average pay for this job?

Try it.

Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?

Solution

$110,000

Try it.

Jake’s water bill is $24.80 per month plus $2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than $60?

Try it.

Kiyoshi’s phone plan costs $17.50 per month plus $0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than $56.60?

Solution

260 messages

Try it.

Marlon’s TV plan costs $49.99 per month plus $5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of $100?

Try it.

Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 for the banquet room plus $32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be $1,500?

Solution

35 people

Try it.

Moshde runs a hairstyling business from her house. She charges $45 for a haircut and style. Her monthly expenses are $960. She wants to be able to put at least $1,200 per month into her savings account order to open her own salon. How many “cut & styles” must she do to save at least $1,200 per month?

Try it.

Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?

Solution

32 jobs

Try it.

Katherine is a personal chef. She charges $115 per four-person meal. Her monthly expenses are $3,150. How many four-person meals must she sell in order to make a profit of at least $1,900?

Try it.

Melissa makes necklaces and sells them online. She charges $88 per necklace. Her monthly expenses are $3,745. How many necklaces must she sell if she wants to make a profit of at least $1,650?

Solution

62 necklaces

Try it.

Five student government officers want to go to the state convention. It will cost them $110 for registration, $375 for transportation and food, and $42 per person for the hotel. There is $450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge $5 per car, how many cars must they wash in order to have enough money to pay for the trip?

Try it.

Cesar is planning a four-day trip to visit his friend at a college in another state. It will cost him $198 for airfare, $56 for local transportation, and $45 per day for food. He has $189 in savings and can earn $35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?

Solution

seven lawns

Try it.

Alonzo works as a car detailer. He charges $175 per car. He is planning to move out of his parents’ house and rent his first apartment. He will need to pay $120 for application fees, $950 for security deposit, and first and last months’ rent at $1,140 per month. He has $1,810 in savings. How many cars must he detail to have enough money to rent the apartment?

Try it.

Eun-Kyung works as a tutor and earns $60 per hour. She has $792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her $1,520 for food and drinks and $150 for the photographer. She will also have a favor for each of the guests, and each favor will cost $7.50. How many hours must she tutor to have enough money for the party?

Solution

20 hours

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality \(x>3\) true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality \(x>3\).

We show the solutions to the inequality \(x>3\) on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of \(x>3\) is shown in . Please note that the following convention is used: thick arrows point in the positive direction and thin arrows point in the negative direction.

The graph of the inequality \(x\ge 3\) is very much like the graph of \(x>3\), but now we need to show that 3 is a solution, too. We do that by putting a bracket at \(x=3\), as shown in .

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

Example

Try it.

Graph on the number line:

ⓐ \(x\le 1\) ⓑ \(x<5\) ⓒ \(x>-1\)

Solution

1. ⓐ \(x\le 1\)
   This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at \(x=1\) to show that it is included.
   
2. ⓑ \(x<5\)
   This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at \(x=5\) to show it is not included.
   
3. ⓒ \(x>-1\)
   This means all numbers greater than \(-1\), but not including \(-1\). We shade in all the numbers on the number line to the right of \(-1\), then put a parenthesis at \(x=-1\) to show it is not included.
   

We can also represent inequalities using interval notation. As we saw above, the inequality \(x>3\) means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express \(x>3\) as \((3,\infty ).\) The symbol \(\infty\) is read as ‘infinity’. It is not an actual number. shows both the number line and the interval notation.

The inequality \(x\le 1\) means all numbers less than or equal to 1. There is no lower end to those numbers. We write \(x\le 1\) in interval notation as \((\text{-}\infty ,1]\). The symbol \(\text{-}\infty\) is read as ‘negative infinity’. shows both the number line and interval notation.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

Similar properties hold true for inequalities.

For example, we know that −4 is less than 2.
If we subtract 5 from both quantities, is the left side still less than the right side?
We get −9 on the left and −3 on the right.
And we know −9 is less than −3.
	The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality \(x+5>9\), the steps would look like this:

	\(x+5>9\)
Subtract 5 from both sides to isolate \(x\).	\(x+5-5>9-5\)
Simplify.	\(x>4\)

Any number greater than 4 is a solution to this inequality.

Example

Try it.

Solve the inequality \(n-\frac{1}{2}\le \frac{5}{8}\), graph the solution on the number line, and write the solution in interval notation.

Solution

Add \(\frac{1}{2}\) to both sides of the inequality.
Simplify.
Graph the solution on the number line.
Write the solution in interval notation.	\((-\infty ,\frac{9}{8}]\)

Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider some numerical examples.

Divide both sides by 5.		Multiply both sides by 5.
Simplify.
Fill in the inequality signs.

\[\text{The inequality signs stayed the same.}\]

Does the inequality stay the same when we divide or multiply by a negative number?

Divide both sides by −5.		Multiply both sides by −5.
Simplify.
Fill in the inequality signs.

\[\text{The inequality signs reversed their direction.}\]

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Here are the Division and Multiplication Properties of Inequality for easy reference.

Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

Example

Try it.

Solve the inequality \(4m\le 9m+17\), graph the solution on the number line, and write the solution in interval notation.

Solution

Subtract \(9m\) from both sides to collect the variables on the left.
Simplify.
Divide both sides of the inequality by −5, and reverse the inequality.
Simplify.
Graph the solution on the number line.
Write the solution in interval notation.

Example

Try it.

Solve the inequality \(8p+3(p-12)>7p-28\), graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible.	\(8p+3(p-12)>7p-28\)
Distribute.	\(\ 8p+3p-36>7p-28\)
Combine like terms.	\(\ 11p-36>7p-28\)
Subtract \(7p\) from both sides to collect the variables on the left.	\(\ 11p-36-7p>7p-28-7p\)
Simplify.	\(\ 4p-36>-28\)
Add 36 to both sides to collect the constants on the right.	\(\ 4p-36+36>-28+36\)
Simplify.	\(\ 4p>8\)
Divide both sides of the inequality by 4; the inequality stays the same.	\(\ \frac{4p}{4}>\frac{8}{4}\)
Simplify.	\(\ p>2\)
Graph the solution on the number line.
Write the solution in interal notation.	\((2,\infty )\)

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

Example

Try it.

Solve the inequality \(8x-2(5-x)<4(x+9)+6x\), graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible.	\(8x-2(5-x)<4(x+9)+6x\)
Distribute.	\(8x-10+2x<4x+36+6x\)
Combine like terms.	\(10x-10<10x+36\)
Subtract \(10x\) from both sides to collect the variables on the left.	\(10x-10-10x<10x+36-10x\)
Simplify.	\(-10<36\\)
The \(x\)’s are gone, and we have a true statement.	The inequality is an identity. The solution is all real numbers.
Graph the solution on the number line.
Write the solution in interval notation.	\((-\infty ,\infty )\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.

shows some common phrases that indicate inequalities.

\(>\)	\(\ge\)	\(<\)	\(\le\)
is greater than	is greater than or equal to	is less than	is less than or equal to
is more than	is at least	is smaller than	is at most
is larger than	is no less than	has fewer than	is no more than
exceeds	is the minimum	is lower than	is the maximum

Example

Try it.

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twelve times c is no more than 96.

Solution

Translate.
Solve—divide both sides by 12.
Simplify.
Write in interval notation.
Graph on the number line.

Example

Try it.

Translate and solve. Then write the solution in interval notation and graph on the number line.

Thirty less than x is at least 45.

Solution

Translate.
Solve—add 30 to both sides.
Simplify.
Write in interval notation.
Graph on the number line.

Key Concepts

• Subtraction Property of Inequality
  For any numbers a, b, and c,
  if \(a if \(a>b\) then \(a-c>b-c.\)
• Addition Property of Inequality
  For any numbers a, b, and c,
  if \(a if \(a>b\) then \(a+c>b+c.\)
• Division and Multiplication Properties of Inequality
  For any numbers a, b, and c,
  if \(a0\), then \(\frac{a}{c}<\frac{b}{c}\) and \(ac>bc\).
  if \(a>b\) and \(c>0\), then \(\frac{a}{c}>\frac{b}{c}\) and \(ac>bc\).
  if \(a\frac{b}{c}\) and \(ac>bc\).
  if \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\) and \(ac
• When we divide or multiply an inequality by a:
  • positive number, the inequality stays the same.
  • negative number, the inequality reverses.

Chapter 2 Review Exercises

Verify a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

Try it.

\(10x-1=5x;x=\frac{1}{5}\)

Try it.

\(w+2=\frac{5}{8};w=\frac{3}{8}\)

Solution

no

Try it.

\(-12n+5=8n;n=-\frac{5}{4}\)

Try it.

\(6a-3=-7a,a=\frac{3}{13}\)

Solution

yes

Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, solve each equation using the Subtraction Property of Equality.

Try it.

\(x+7=19\)

Try it.

\(y+2=-6\)

Solution

\(y=-8\)

Try it.

\(a+\frac{1}{3}=\frac{5}{3}\)

Try it.

\(n+3.6=5.1\)

Solution

\(n=1.5\)

In the following exercises, solve each equation using the Addition Property of Equality.

Try it.

\(u-7=10\)

Try it.

\(x-9=-4\)

Solution

\(x=5\)

Try it.

\(c-\frac{3}{11}=\frac{9}{11}\)

Try it.

\(p-4.8=14\)

Solution

\(p=18.8\)

In the following exercises, solve each equation.

Try it.

\(n-12=32\)

Try it.

\(y+16=-9\)

Solution

\(y=-25\)

Try it.

\(f+\frac{2}{3}=4\)

Try it.

\(d-3.9=8.2\)

Solution

\(d=12.1\)

Solve Equations That Require Simplification

In the following exercises, solve each equation.

Try it.

\(y+8-15=-3\)

Try it.

\(7x+10-6x+3=5\)

Solution

\(x=-8\)

Try it.

\(6(n-1)-5n=-14\)

Try it.

\(8(3p+5)-23(p-1)=35\)

Solution

\(p=-28\)

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Try it.

The sum of \(-6\) and \(m\) is 25.

Try it.

Four less than \(n\) is 13.

Solution

\(n-4=13;n=17\)

Translate and Solve Applications

In the following exercises, translate into an algebraic equation and solve.

Try it.

Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?

Try it.

Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?

Solution

161 pounds

Try it.

Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?

Try it.

Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

Solution

$131.19

Solve an Equation with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

Try it.

\(8p+7=47\)

Try it.

\(10w-5=65\)

Solution

\(w=7\)

Try it.

\(3x+19=-47\)

Try it.

\(32=-4-9n\)

Solution

\(n=-4\)

Solve an Equation with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

Try it.

\(7y=6y-13\)

Try it.

\(5a+21=2a\)

Solution

\(a=-7\)

Try it.

\(k=-6k-35\)

Try it.

\(4x-\frac{3}{8}=3x\)

Solution

\(x=\frac{3}{8}\)

Solve an Equation with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

Try it.

\(12x-9=3x+45\)

Try it.

\(5n-20=-7n-80\)

Solution

\(n=-5\)

Try it.

\(4u+16=-19-u\)

Try it.

\(\frac{5}{8}c-4=\frac{3}{8}c+4\)

Solution

\(c=32\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Chapter 2 Practice Test

In the following exercises, solve each equation.

In the following exercises, graph on the number line and write in interval notation.

In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

In the following exercises, translate to an equation or inequality and solve.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Translate from algebra to English: \(15>x.\)
   If you missed this problem, review .

   జవాబు వెల్లడి చేయండి

   \(15\) is greater than \(x\).

2. Translate to an algebraic expression: 15 is less than x.
   If you missed this problem, review .

   జవాబు వెల్లడి చేయండి

   \(15

3. Graph each inequality on the number line and write in interval notation.

   ⓐ \(x\ge -3\) ⓑ \(x<2.5\) ⓒ \(x\le -\frac{3}{5}\)

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   Shade to the right of \(-3,\) and put a bracket at \(-3.\)
   Write in interval notation.

   ⓑ
   

   Shade to the left of 2.5 and put a parenthesis at 2.5.
   Write in interval notation.

   ⓒ
   

   Shade to the left of \(-\frac{3}{5},\) and put a bracket at \(-\frac{3}{5}.\)
   Write in interval notation.

4. Graph each inequality on the number line and write in interval notation: ⓐ \(x>2\) ⓑ \(x\le -1.5\) ⓒ \(x\ge \frac{3}{4}.\)

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   ⓑ
   

   ⓒ
   

5. Graph each inequality on the number line and write in interval notation: ⓐ \(x\le -4\) ⓑ \(x\ge 0.5\) ⓒ \(x<-\frac{2}{3}.\)

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   ⓑ
   

   ⓒ
   

6. Graph each inequality on the number line and write in interval notation.

   ⓐ \(-3

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   Shade between \(-3\) and 4. Put a parentheses at \(-3\) and 4.
   Write in interval notation.

   ⓑ
   

   Shade between \(-6\) and −1. Put a bracket at \(-6,\) and a parenthesis at −1.
   Write in interval notation.

   ⓒ
   

   Shade between 0 and 2.5. Put a bracket at 0 and at 2.5.
   Write in interval notation.

7. Graph each inequality on the number line and write in interval notation:

   ⓐ \(-2

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   ⓑ
   

   ⓒ
   

8. Graph each inequality on the number line and write in interval notation:

   ⓐ \(-6

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   ⓑ
   

   ⓒ
   

9. Solve each inequality. Graph the solution on the number line, and write the solution in interval notation.

   ⓐ \(x-\frac{3}{8}\le \frac{3}{4}\) ⓑ \(9y<\text{}\text{}54\) ⓒ \(-15<\frac{3}{5}z\)

   జవాబు వెల్లడి చేయండి

   ⓐ
   

   Add \(\frac{3}{8}\) to both sides of the inequality.
   Simplify.
   Graph the solution on the number line.
   Write the solution in interval notation.

   ⓑ
   

   Divide both sides of the inequality by 9; since   9 is positive, the inequality stays the same.
   Simplify.
   Graph the solution on the number line.
   Write the solution in interval notation.

   ⓒ
   

   Multiply both sides of the inequality by \(\frac{5}{3}.\) Since \(\frac{5}{3}\) is positive, the inequality stays the same.
   Simplify.
   Rewrite with the variable on the left.
   Graph the solution on the number line.
   Write the solution in interval notation.

10. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

    ⓐ \(p-\frac{3}{4}\ge \frac{1}{6}\) ⓑ \(9c>72\) ⓒ \(24\le \frac{3}{8}m\)

    జవాబు వెల్లడి చేయండి

    ⓐ
    

    ⓑ
    

    ⓒ
    

11. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

    ⓐ \(r-\frac{1}{3}\le \frac{7}{12}\) ⓑ \(12d\le \text{}60\) ⓒ \(-24<\frac{4}{3}n\)

    జవాబు వెల్లడి చేయండి

    ⓐ
    

    ⓑ
    

    ⓒ
    

12. Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    ⓐ \(-13m\ge 65\) ⓑ \(\frac{n}{-2}\ge 8\)

    జవాబు వెల్లడి చేయండి

    ⓐ
    

    Divide both sides of the inequality by \(-13.\) Since \(-13\) is a negative, the inequality reverses.
    Simplify.
    Graph the solution on the number line.
    Write the solution in interval notation.

    ⓑ
    

    Multiply both sides of the inequality by \(-2.\) Since \(-2\) is a negative, the inequality reverses.
    Simplify.
    Graph the solution on the number line.
    Write the solution in interval notation.

13. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

    ⓐ \(-8q<32\) ⓑ \(\frac{k}{-12}\le 15.\)

    జవాబు వెల్లడి చేయండి

    ⓐ
    

    ⓑ
    

14. Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

    ⓐ \(-7r\le \text{}-70\) ⓑ \(\frac{u}{-4}\ge -16.\)

    జవాబు వెల్లడి చేయండి

    ⓐ
    

    ⓑ
    

15. Solve the inequality \(6y\le 11y+17,\) graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

    Subtract \(11y\) from both sides to collect the variables on the left.
    Simplify.
    Divide both sides of the inequality by \(-5,\) and reverse the inequality.
    Simplify.
    Graph the solution on the number line.
    Write the solution in interval notation.

16. Solve the inequality, graph the solution on the number line, and write the solution in interval notation: \(3q\ge 7q-23.\)

17. Solve the inequality, graph the solution on the number line, and write the solution in interval notation: \(6x<10x+19.\)

18. Solve the inequality \(8p+3(p-12)>7p-28,\) graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

    	\(8p+3(p-12)>7p-28\)
    Simplify each side as much as possible.
    Distribute.	\(\ 8p+3p-36>7p-28\)
    Combine like terms.	\(\ 11p-36>7p-28\)
    Subtract \(7p\) from both sides to collect the variables on the left, since \(11>7.\)	\(\ 11p-36-7p>7p-28-7p\)
    Simplify.	\(\ 4p-36>-28\)
    Add 36 to both sides to collect the constants on the right.	\(\ 4p-36+36>-28+36\)
    Simplify.	\(\ 4p>8\)
    Divide both sides of the inequality by 4; the inequality stays the same.	\(\ \frac{4p}{4}>\text{}\frac{8}{4}\)
    Simplify.	\(\ p>2\)
    Graph the solution on the number line.
    Write the solution in interval notation.	\((2,\infty )\\)

19. Solve the inequality \(9y+2(y+6)>5y-24\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

20. Solve the inequality \(6u+8(u-1)>10u+32\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

21. Solve the inequality \(8x-2(5-x)<4(x+9)+6x,\) graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

    Simplify each side as much as possible.	\(\\)	\(8x-2(5-x)<4(x+9)+6x\)
    Distribute.		\(8x-10+2x<4x+36+6x\)
    Combine like terms.		\(10x-10<10x+36\)
    Subtract 10x from both sides to collect the variables on the left.		\(10x-10-10x<10x+36-10x\)
    Simplify.		\(-10<36\\)
    The x’s are gone, and we have a true statement.		The inequality is an identity. The solution is all real numbers.
    Graph the solution on the number line.
    Write the solution in interval notation.		\((\text{-}\infty ,\infty )\)

22. Solve the inequality \(4b-3(3-b)>5(b-6)+2b\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

23. Solve the inequality \(9h-7(2-h)<8(h+11)+8h\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

24. Solve the inequality \(\frac{1}{3}a-\frac{1}{8}a>\frac{5}{24}a\text{}+\frac{3}{4},\) graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

    Multiply both sides by the LCD, 24, to clear the fractions.
    Simplify.
    Combine like terms.
    Subtract \(5a\) from both sides to collect the variables on the left.
    Simplify.
    The statement is false.	The inequality is a contradiction. There is no solution.
    Graph the solution on the number line.
    Write the solution in interval notation.	There is no solution.

25. Solve the inequality \(\frac{1}{4}x-\frac{1}{12}x>\frac{1}{6}x+\frac{7}{8}\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

26. Solve the inequality \(\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z\text{}+\frac{3}{5}\), graph the solution on the number line, and write the solution in interval notation.

    జవాబు వెల్లడి చేయండి

27. Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

    \[\text{Twenty-seven less than}\ x\ \text{is at least 48.}\]
    జవాబు వెల్లడి చేయండి

    Translate.
    Solve—add 27 to both sides.
    Simplify.
    Graph on the number line.
    Write in interval notation.

28. Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

    Nineteen less than p is no less than 47.

    జవాబు వెల్లడి చేయండి

29. Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

    Four more than a is at most 15.

    జవాబు వెల్లడి చేయండి

30. Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?

    జవాబు వెల్లడి చేయండి

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	the maximum number of tablets Dawn can buy
    Step 3. Name what you are looking for.
    Choose a variable to represent that quantity.	\(\text{Let}\ n=\text{the number of tablets.}\)
    Step 4. Translate. Write a sentence that gives the information to find it.	$254.12 times the number of tablets is no more than $4,000.
    Translate into an inequality.	\(254.12n\le 4000\)
    Step 5. Solve the inequality. But n must be a whole number of tablets, so round to 15.	\(\begin{array}{l} \\ \\ n\le 15.74 \\ n\le 15\end{array}\)
    Step 6. Check the answer in the problem and make sure it makes sense.
    Rounding down the price to $250, 15 tablets would cost $3,750, while 16 tablets would be $4,000. So a maximum of 15 tablets at $254.12 seems reasonable.
    Step 7. Answer the question with a complete sentence.	Dawn can buy a maximum of 15 tablets.

31. Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. What is the maximum number of packs she can buy?

    జవాబు వెల్లడి చేయండి

    Angie can buy 7 packs of juice.

32. Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?

    జవాబు వెల్లడి చేయండి

    Daniel can have 11 people at the party.

33. Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. How many text messages can she send/receive and keep her monthly phone bill no more than $50?

    జవాబు వెల్లడి చేయండి

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	the number of text messages Taleisha can make
    Step 3. Name what you are looking for.
    Choose a variable to represent that quantity.	\(\text{Let}\ t=\text{the number of text messages.}\)
    Step 4. Translate Write a sentence that gives the information to find it.	$28.80 plus $0.20 times the number of text messages is less than or equal to $50.
    Translate into an inequality.	\(\ 28.80+0.20t\le 50\)
    Step 5. Solve the inequality.	\(\begin{array}{l} \\ \\ 0.2t\le 21.2 \\ \\ t\le 106\ \text{text messages}\end{array}\)
    Step 6. Check the answer in the problem and make sure it makes sense. \(\text{Yes,}\ 28.80+0.20(106)=50.\)
    Step 7. Write a sentence that answers the question.	Taleisha can send/receive no more than 106 text messages to keep her bill no more than $50.

34. Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can they travel during the week and still keep within their $200 budget?

    జవాబు వెల్లడి చేయండి

    Sergio and Lizeth can travel no more than 500 miles.

35. Rameen’s heating bill is $5.42 per month plus $1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of $87.50.

    జవాబు వెల్లడి చేయండి

    Rameen can use no more than 76 therms.

36. Felicity has a calligraphy business. She charges $2.50 per wedding invitation. Her monthly expenses are $650. How many invitations must she write to earn a profit of at least $2,800 per month?

    జవాబు వెల్లడి చేయండి

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	the number of invitations Felicity needs to write
    Step 3. Name what you are looking for. Choose a variable to represent it.	\(\text{Let}\ j=\text{the number of invitations.}\)
    Step 4. Translate. Write a sentence that gives the information to find it.	$2.50 times the number of invitations minus $650 is at least $2,800.
    Translate into an inequality.	\(2.50j-650\ge 2,800\)
    Step 5. Solve the inequality.	\(\begin{array}{l} \\ 2.5j\ge 3,450 \\ j\ge 1,380\ \text{invitations}\end{array}\)
    Step 6. Check the answer in the problem and make sure it makes sense.
    If Felicity wrote 1400 invitations, her profit would be 2.50(1400) − 650, or $2,850. This is more than $2800.
    Step 7. Write a sentence that answers the question.	Felicity must write at least 1,380 invitations.

37. Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are $2,272. How many hours must he work in order to earn a profit of at least $800 per month?

    జవాబు వెల్లడి చేయండి

    Caleb must work at least 96 hours.

38. Elliot has a landscape maintenance business. His monthly expenses are $1,100. If he charges $60 per job, how many jobs must he do to earn a profit of at least $4,000 a month?

    జవాబు వెల్లడి చేయండి

    Elliot must work at least 85 jobs.

39. Malik is planning a six-day summer vacation trip. He has $840 in savings, and he earns $45 per hour for tutoring. The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?

    జవాబు వెల్లడి చేయండి

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.	the number of hours Malik must tutor
    Step 3. Name what you are looking for.
    Choose a variable to represent that quantity.	\(\text{Let}\ h=\text{the number of hours.}\)
    Step 4. Translate. Write a sentence that gives the information to find it.	The expenses must be less than or equal to the income. The cost of airfare plus the cost of food and sightseeing and the hotel bill must be less than the savings plus the amount earned tutoring.
    Translate into an inequality.	\(525+780+95(6)\le 840+45h\)
    Step 5. Solve the inequality.	\(\ \begin{array}{l}1,875\le 840+45h \\ 1,035\le 45h \\ 23\le h \\ h\ge 23\end{array}\)
    Step 6. Check the answer in the problem and make sure it makes sense. We substitute 23 into the inequality. \(\begin{array}{l} \\ \\ 1,875\le 840+45h \\ 1,875\le 840+45(23) \\ 1,875\le 1875\end{array}\)
    Step 7. Write a sentence that answers the question.	Malik must tutor at least 23 hours.

40. Brenda’s best friend is having a destination wedding and the event will require 3 nights in a hotel. Brenda has $500 in savings and can earn $15 an hour babysitting. She expects to pay $350 airfare, $375 for food and entertainment and $60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?

    జవాబు వెల్లడి చేయండి

    Brenda must babysit at least 27 hours.

Symbols used here

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solve Linear Inequalities

1. Graph inequalities on the number line
2. Solve linear inequalities
3. Translate words to an inequality and solve
4. Solve applications with linear inequalities
5. positive number, the inequality stays the same.
6. negative number, the inequality reverses.
7. A
8. For any numbers

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

మీ సొంత ప్రయత్నించండి

⌨ Type ✎ Draw

మీ గణిత సమస్య

పరిష్కరించు

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

ఇంకా Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value