Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form \(x+a=b\) or \(x-a=b\). We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Let’s look at our puzzle again with the envelopes and counters in .

In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since \(6\div 2=3\)).

What equation models the situation shown in ? There are two envelopes, and each contains \(x\) counters. Together, the two envelopes must contain a total of 6 counters.

If we divide both sides of the equation by 2, as we did with the envelopes and counters,
we get:

We found that each envelope contains 3 counters. Does this check? We know \(2\cdot 3=6\), so it works! Three counters in each of two envelopes does equal six!

This example leads to the Division Property of Equality.

Example

Try it.

Solve: \(5x=-27.\)

Solution

To isolate \(x\), “undo” the multiplication by 5.
Divide to ‘undo’ the multiplication.
Simplify.
Check:
Substitute \(-\frac{27}{5}\) for \(x.\)
		Since this is a true statement, \(x=-\frac{27}{5}\) is the solution to \(5x=-27\).

Example

Try it.

Solve: \(\frac{y}{-7}=-14.\)

Solution

Here \(y\) is divided by \(-7\). We must multiply by \(-7\) to isolate \(y\).

Multiply both sides by \(-7\).
Multiply.
Simplify.
Check: \(\frac{y}{-7}=-14\)
Substitute \(y=98\).
Divide.

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

Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Example

Try it.

Solve: \(14-23=12y-4y-5y.\)

Solution

Begin by simplifying each side of the equation.

Simplify each side.
Divide both sides by \(3\).
Check:
Substitute \(y=-3\).

Example

Try it.

Solve: \(-4(a-3)-7=25.\)

Solution

Here we will simplify each side of the equation by using the distributive property first.

Distribute.
Simplify.
Simplify.
Divide both sides by \(-4\) to isolate \(a\).
Divide.
Check:
Substitute \(a=-5\).

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Example

Try it.

Translate and solve: The number 143 is the product of \(-11\) and y.

Solution

Begin by translating the sentence into an equation.

Translate.
Divide by \(-11\).
Simplify.
Check: \(\begin{array}{llll} & 143 & = & -11y \\ & 143 & \overset{?}{=} & -11(-13) \\ & 143 & = & 143✓\end{array}\)

Example

Try it.

Translate and solve: \(n\) divided by 8 is \(-32\).

Solution

Begin by translating the sentence into an equation. Translate.
Multiple both sides by 8.
Simplify.
Check:	Is \(n\) divided by 8 equal to −32?
Let \(n=-256\).	Is \(-256\) divided by \(8\) equal to \(-32\)?
Translate.	\(\frac{-256}{8}\overset{?}{=}-32\)
Simplify.	\(\ -32=-32✓\)

Example

Try it.

Translate and solve: The quotient of \(y\) and \(-4\) is \(68\).

Solution

Begin by translating the sentence into an equation.

Translate.
Multiply both sides by \(-4\).
Simplify.
Check:	Is the quotient of \(y\) and \(-4\) equal to \(68\)?
Let \(y=-272\).	Is the quotient of \(-272\) and \(-4\) equal to \(68\)?
Translate.	\(\frac{-272}{-4}\overset{?}{=}68\)
Simplify.	\(\ 68=68✓\)

Example

Try it.

Translate and solve: Three-fourths of \(p\) is 18.

Solution

Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

Translate.
Multiply both sides by \(\frac{4}{3}.\)
Simplify.
Check:	Is three-fourths of p equal to 18?
Let \(p=24.\)	Is three-fourths of 24 equal to 18?
Translate.	\(\frac{3}{4}\cdot \ 24\ \overset{?}{=}18\)
Simplify.	\(\ 18=18✓\)

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

Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

Example

Try it.

Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

Solution

What are you asked to find?	The cost of 1 pound of grapes
Assign a variable.	Let \(c\) = the cost of one pound.
Write a sentence that gives the information to find it.	The cost of 6 pounds is $10.74.
Translate into an equation.	\(6c=10.74\)
Solve.	\(\begin{array}{l}\frac{6c}{6}=\frac{10.74}{6} \\ c=1.79\end{array}\)
	The grapes cost $1.79 per pound.
Check: If one pound costs $1.79, do 6 pounds cost #10.74? \(\begin{array}{lll}6(1.79) & \overset{?}{=} & 10.74 \\ 10.74 & = & 10.74✓\end{array}\)

Example

Try it.

Andreas bought a used car for $12,000. Because the car was 4-years old, its price was \(\frac{3}{4}\) of the original price, when the car was new. What was the original price of the car?

Solution

What are you asked to find?	The original price of the car
Assign a variable.	Let \(p\) = the original price.
Write a sentence that gives the information to find it.	$12,000 is \(\frac{3}{4}\) of the original price.
Translate into an equation.	\(12,000=\frac{3}{4}p\)
Solve.	\(\begin{array}{l}\frac{4}{3}(12,000)=\frac{4}{3}\cdot \frac{3}{4}p \\ 16,000=p\end{array}\)
	The original cost of the car was $16,000.
Check: Is \(\frac{3}{4}\) of $16,000 equal to $12,000? \(\begin{array}{lll}\frac{3}{4}\cdot 16,000 & \overset{?}{=} & 12,000 \\ 12,000 & = & 12,000✓\end{array}\)

Key Concepts

• The Division Property of Equality—For any numbers a, b, and c, and \(c\ne 0\), if \(a=b\), then \(\frac{a}{c}=\frac{b}{c}\).
  When you divide both sides of an equation by any non-zero number, you still have equality.
• The Multiplication Property of Equality—For any numbers a, b, and c, if \(a=b\), then \(ac=bc\).
  If you multiply both sides of an equation by the same number, you still have equality.

Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

Try it.

\(8x=56\)

Solution

\(x=7\)

Try it.

\(7p=63\)

Try it.

\(-5c=55\)

Solution

\(c=-11\)

Try it.

\(-9x=-27\)

Try it.

\(-809=15y\)

Solution

\(y=-\frac{809}{15}\)

Try it.

\(-731=19y\)

Try it.

\(-37p=-541\)

Solution

\(p=\frac{541}{37}\)

Try it.

\(-19m=-586\)

Try it.

\(0.25z=3.25\)

Solution

\(z=13\)

Try it.

\(0.75a=11.25\)

Try it.

\(-13x=0\)

Solution

\(x=0\)

Try it.

\(24x=0\)

Try it.

\(\frac{x}{4}=35\)

Solution

\(x=140\)

Try it.

\(\frac{z}{2}=54\)

Try it.

\(-20=\frac{q}{-5}\)

Solution

\(q=100\)

Try it.

\(\frac{c}{-3}=-12\)

Try it.

\(\frac{y}{9}=-16\)

Solution

\(y=-144\)

Try it.

\(\frac{q}{6}=-38\)

Try it.

\(\frac{m}{-12}=45\)

Solution

\(m=-540\)

Try it.

\(-24=\frac{p}{-20}\)

Try it.

\(\text{-}y=6\)

Solution

\(y=-6\)

Try it.

\(\text{-}u=15\)

Try it.

\(\text{-}v=-72\)

Solution

\(v=72\)

Try it.

\(\text{-}x=-39\)

Try it.

\(\frac{2}{3}y=48\)

Solution

\(y=72\)

Try it.

\(\frac{3}{5}r=75\)

Try it.

\(-\frac{5}{8}w=40\)

Solution

\(w=-64\)

Try it.

\(24=-\frac{3}{4}x\)

Try it.

\(-\frac{2}{5}=\frac{1}{10}a\)

Solution

\(a=-4\)

Try it.

\(-\frac{1}{3}q=-\frac{5}{6}\)

Try it.

\(-\frac{7}{10}x=-\frac{14}{3}\)

Solution

\(x=\frac{20}{3}\)

Try it.

\(\frac{3}{8}y=-\frac{1}{4}\)

Try it.

\(\frac{7}{12}=-\frac{3}{4}p\)

Solution

\(p=-\frac{7}{9}\)

Try it.

\(\frac{11}{18}=-\frac{5}{6}q\)

Try it.

\(-\frac{5}{18}=-\frac{10}{9}u\)

Solution

\(u=\frac{1}{4}\)

Try it.

\(-\frac{7}{20}=-\frac{7}{4}v\)

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Try it.

\(100-16=4p-10p-p\)

Solution

\(p=-12\)

Try it.

\(-18-7=5t-9t-6t\)

Try it.

\(\frac{7}{8}n-\frac{3}{4}n=9+2\)

Solution

\(n=88\)

Try it.

\(\frac{5}{12}q+\frac{1}{2}q=25-3\)

Try it.

\(0.25d+0.10d=6-0.75\)

Solution

\(d=15\)

Try it.

\(0.05p-0.01p=2+0.24\)

Try it.

\(-10(q-4)-57=93\)

Solution

\(q=-11\)

Try it.

\(-12(d-5)-29=43\)

Try it.

\(-10(x+4)-19=85\)

Solution

\(x=-\frac{72}{5}\)

Try it.

\(-15(z+9)-11=75\)

Mixed Practice

In the following exercises, solve each equation.

Try it.

\(\frac{9}{10}x=90\)

Solution

\(x=100\)

Try it.

\(\frac{5}{12}y=60\)

Try it.

\(y+46=55\)

Solution

\(y=9\)

Try it.

\(x+33=41\)

Try it.

\(\frac{w}{-2}=99\)

Solution

\(w=-198\)

Try it.

\(\frac{s}{-3}=-60\)

Try it.

\(27=6a\)

Solution

\(a=\frac{9}{2}\)

Try it.

\(\text{-}a=7\)

Try it.

\(\text{-}x=2\)

Solution

\(x=-2\)

Try it.

\(z-16=-59\)

Try it.

\(m-41=-14\)

Solution

\(m=27\)

Try it.

\(0.04r=52.60\)

Try it.

\(63.90=0.03p\)

Solution

\(p=2130\)

Try it.

\(-15x=-120\)

Try it.

\(84=-12z\)

Solution

\(z=-7\)

Try it.

\(19.36=x-0.2x\)

Try it.

\(c-0.3c=35.70\)

Solution

\(c=51\)

Try it.

\(\text{-}y=-9\)

Try it.

\(\text{-}x=-8\)

Solution

\(x=8\)

Translate to an Equation and Solve

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

Try it.

187 is the product of \(-17\) and m.

Try it.

133 is the product of \(-19\) and n.

Solution

\(133=-19n;n=-7\)

Try it.

\(-184\) is the product of 23 and p.

Try it.

\(-152\) is the product of 8 and q.

Solution

\(-152=8q;q=-19\)

Try it.

u divided by 7 is equal to \(-49\).

Try it.

r divided by 12 is equal to \(-48\).

Solution

\(\frac{r}{12}=-48;r=-576\)

Try it.

h divided by \(-13\) is equal to \(-65\).

Try it.

j divided by \(-20\) is equal to \(-80\).

Solution

\(\frac{j}{-20}=-80;j=1,600\)

Try it.

The quotient \(c\) and \(-19\) is 38.

Try it.

The quotient of \(b\) and \(-6\) is 18.

Solution

\(\frac{b}{-6}=18;b=-108\)

Try it.

The quotient of \(h\) and 26 is \(-52\).

Try it.

The quotient \(k\) and 22 is \(-66\).

Solution

\(\frac{k}{22}=-66;k=-1,452\)

Try it.

Five-sixths of y is 15.

Try it.

Three-tenths of x is 15.

Solution

\(\frac{3}{10}x=15;x=50\)

Try it.

Four-thirds of w is 36.

Try it.

Five-halves of v is 50.

Solution

\(\frac{5}{2}v=50;v=20\)

Try it.

The sum of nine-tenths and g is two-thirds.

Try it.

The sum of two-fifths and f is one-half.

Solution

\(\frac{2}{5}+f=\frac{1}{2};f=\frac{1}{10}\)

Try it.

The difference of p and one-sixth is two-thirds.

Try it.

The difference of q and one-eighth is three-fourths.

Solution

\(q-\frac{1}{8}=\frac{3}{4};q=\frac{7}{8}\)

Translate and Solve Applications

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

Try it.

Kindergarten Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?

Try it.

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

Solution

6 balloons

Try it.

Tickets Mollie paid $36.25 for 5 movie tickets. What was the price of each ticket?

Try it.

Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

Solution

$1.08

Try it.

Sewing Nancy used 14 yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

Try it.

MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

Solution

36 mpg

Try it.

Height Aiden is 27 inches tall. He is \(\frac{3}{8}\) as tall as his father. How tall is his father?

Try it.

Real estate Bea earned $11,700 commission for selling a house, calculated as \(\frac{6}{100}\) of the selling price. What was the selling price of the house?

Solution

$195,000

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

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. Simplify: \(-7(\frac{1}{-7}).\)
   If you missed this problem, review .

   Cevabı açıkla.

   \(1\)

2. Evaluate \(9x+2\) when \(x=-3\).
   If you missed this problem, review .

   Cevabı açıkla.

   \(-25\)

3. Solve: \(5x=-27.\)

   Cevabı açıkla.

   To isolate \(x\), “undo” the multiplication by 5.
   Divide to ‘undo’ the multiplication.
   Simplify.
   Check:
   Substitute \(-\frac{27}{5}\) for \(x.\)
   		Since this is a true statement, \(x=-\frac{27}{5}\) is the solution to \(5x=-27\).

4. Solve: \(3y=-41.\)

   Cevabı açıkla.

   \(y=\frac{-41}{3}\)

5. Solve: \(4z=-55.\)

   Cevabı açıkla.

   \(z=\frac{-55}{4}\)

6. Solve: \(\frac{y}{-7}=-14.\)

   Cevabı açıkla.

   Here \(y\) is divided by \(-7\). We must multiply by \(-7\) to isolate \(y\).

   Multiply both sides by \(-7\).
   Multiply.
   Simplify.
   Check: \(\frac{y}{-7}=-14\)
   Substitute \(y=98\).
   Divide.

7. Solve: \(\frac{a}{-7}=-42.\)

   Cevabı açıkla.

   \(a=294\)

8. Solve: \(\frac{b}{-6}=-24.\)

   Cevabı açıkla.

   \(b=144\)

9. Solve: \(\text{-}n=9.\)

   Cevabı açıkla.

   Remember \(-n\) is equivalent to \(-1n\).
   Divide both sides by \(-1\).
   Divide.
   Notice that there are two other ways to solve \(-n=9\). We can also solve this equation by multiplying both sides by \(-1\) and also by taking the opposite of both sides.
   Check:
   Substitute \(n=-9\).
   Simplify.

10. Solve: \(\text{-}k=8.\)

    Cevabı açıkla.

    \(k=-8\)

11. Solve: \(\text{-}g=3.\)

    Cevabı açıkla.

    \(g=-3\)

12. Solve: \(\frac{3}{4}x=12.\)

    Cevabı açıkla.

    Since the product of a number and its reciprocal is 1, our strategy will be to isolate \(x\) by multiplying by the reciprocal of \(\frac{3}{4}\).

    Multiply by the reciprocal of \(\frac{3}{4}\).
    Reciprocals multiply to 1.
    Multiply.
    Notice that we could have divided both sides of the equation \(\frac{3}{4}x=12\) by \(\frac{3}{4}\) to isolate \(x\). While this would work, most people would find multiplying by the reciprocal easier.
    Check:
    Substitute \(x=16\).

13. Solve: \(\frac{2}{5}n=14.\)

    Cevabı açıkla.

    \(n=35\)

14. Solve: \(\frac{5}{6}y=15.\)

    Cevabı açıkla.

    \(y=18\)

15. Solve: \(\frac{8}{15}=-\frac{4}{5}x.\)

    Cevabı açıkla.

    Multiply by the reciprocal of \(-\frac{4}{5}\).
    Reciprocals multiply to 1.
    Multiply.
    Check:
    Let \(x=-\frac{2}{3}\).

16. Solve: \(\frac{9}{25}=-\frac{4}{5}\ z.\)

    Cevabı açıkla.

    \(z=-\frac{9}{20}\)

17. Solve: \(\frac{5}{6}=-\frac{8}{3}\ r.\)

    Cevabı açıkla.

    \(r=-\frac{5}{16}\)

18. Solve: \(14-23=12y-4y-5y.\)

    Cevabı açıkla.

    Begin by simplifying each side of the equation.

    Simplify each side.
    Divide both sides by \(3\).
    Check:
    Substitute \(y=-3\).

19. Solve: \(18-27=15c-9c-3c.\)

    Cevabı açıkla.

    \(c=-3\)

20. Solve:\(18-22=12x-x-4x.\)

    Cevabı açıkla.

    \(x=-\frac{4}{7}\)

21. Solve: \(-4(a-3)-7=25.\)

    Cevabı açıkla.

    Here we will simplify each side of the equation by using the distributive property first.

    Distribute.
    Simplify.
    Simplify.
    Divide both sides by \(-4\) to isolate \(a\).
    Divide.
    Check:
    Substitute \(a=-5\).

22. Solve: \(-4(q-2)-8=24.\)

    Cevabı açıkla.

    \(q=-6\)

23. Solve: \(-6(r-2)-12=30.\)

    Cevabı açıkla.

    \(r=-5\)

24. Translate and solve: The number 143 is the product of \(-11\) and y.

    Cevabı açıkla.

    Begin by translating the sentence into an equation.

    Translate.
    Divide by \(-11\).
    Simplify.
    Check: \(\begin{array}{llll} & 143 & = & -11y \\ & 143 & \overset{?}{=} & -11(-13) \\ & 143 & = & 143✓\end{array}\)

25. Translate and solve: The number 132 is the product of −12 and y.

    Cevabı açıkla.

    \(132=-12y;y=-11\)

26. Translate and solve: The number 117 is the product of −13 and z.

    Cevabı açıkla.

    \(117=-13z;z=-9\)

27. Translate and solve: \(n\) divided by 8 is \(-32\).

    Cevabı açıkla.

    Begin by translating the sentence into an equation. Translate.
    Multiple both sides by 8.
    Simplify.
    Check:	Is \(n\) divided by 8 equal to −32?
    Let \(n=-256\).	Is \(-256\) divided by \(8\) equal to \(-32\)?
    Translate.	\(\frac{-256}{8}\overset{?}{=}-32\)
    Simplify.	\(\ -32=-32✓\)

28. Translate and solve: \(n\) divided by 7 is equal to \(-21\).

    Cevabı açıkla.

    \(\frac{n}{7}=-21;n=-147\)

29. Translate and solve: \(n\) divided by 8 is equal to \(-56\).

    Cevabı açıkla.

    \(\frac{n}{8}=-56;n=-448\)

30. Translate and solve: The quotient of \(y\) and \(-4\) is \(68\).

    Cevabı açıkla.

    Begin by translating the sentence into an equation.

    Translate.
    Multiply both sides by \(-4\).
    Simplify.
    Check:	Is the quotient of \(y\) and \(-4\) equal to \(68\)?
    Let \(y=-272\).	Is the quotient of \(-272\) and \(-4\) equal to \(68\)?
    Translate.	\(\frac{-272}{-4}\overset{?}{=}68\)
    Simplify.	\(\ 68=68✓\)

31. Translate and solve: The quotient of \(q\) and \(-8\) is 72.

    Cevabı açıkla.

    \(\frac{q}{-8}=72;q=-576\)

32. Translate and solve: The quotient of \(p\) and \(-9\) is 81.

    Cevabı açıkla.

    \(\frac{p}{-9}=81;p=-729\)

33. Translate and solve: Three-fourths of \(p\) is 18.

    Cevabı açıkla.

    Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

    Translate.
    Multiply both sides by \(\frac{4}{3}.\)
    Simplify.
    Check:	Is three-fourths of p equal to 18?
    Let \(p=24.\)	Is three-fourths of 24 equal to 18?
    Translate.	\(\frac{3}{4}\cdot \ 24\ \overset{?}{=}18\)
    Simplify.	\(\ 18=18✓\)

34. Translate and solve: Two-fifths of \(f\) is 16.

    Cevabı açıkla.

    \(\frac{2}{5}f=16;f=40\)

35. Translate and solve: Three-fourths of \(f\) is 21.

    Cevabı açıkla.

    \(\frac{3}{4}f=21;f=28\)

36. Translate and solve: The sum of three-eighths and \(x\) is one-half.

    Cevabı açıkla.

    Begin by translating the sentence into an equation.

    Translate.
    Subtract \(\frac{3}{8}\) from each side.
    Simplify and rewrite fractions with common denominators.
    Simplify.
    Check:		Is the sum of three-eighths and \(x\) equal to one-half?
    \(\text{Let}\ x=\frac{1}{8}.\)		Is the sum of three-eighths and one-eighth equal to one-half?
    Translate.		\(\ \frac{3}{8}+\frac{1}{8}\overset{?}{=}\frac{1}{2}\)
    Simplify.		\(\ \frac{4}{8}\overset{?}{=}\frac{1}{2}\)
    Simplify.		\(\ \frac{1}{2}=\frac{1}{2}✓\)

37. Translate and solve: The sum of five-eighths and x is one-fourth.

    Cevabı açıkla.

    \(\frac{5}{8}+x=\frac{1}{4};x=-\frac{3}{8}\)

38. Translate and solve: The sum of three-fourths and x is five-sixths.

    Cevabı açıkla.

    \(\frac{3}{4}+x=\frac{5}{6};x=\frac{1}{12}\)

39. Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

    Cevabı açıkla.

    What are you asked to find?	The cost of 1 pound of grapes
    Assign a variable.	Let \(c\) = the cost of one pound.
    Write a sentence that gives the information to find it.	The cost of 6 pounds is $10.74.
    Translate into an equation.	\(6c=10.74\)
    Solve.	\(\begin{array}{l}\frac{6c}{6}=\frac{10.74}{6} \\ c=1.79\end{array}\)
    	The grapes cost $1.79 per pound.
    Check: If one pound costs $1.79, do 6 pounds cost #10.74? \(\begin{array}{lll}6(1.79) & \overset{?}{=} & 10.74 \\ 10.74 & = & 10.74✓\end{array}\)

40. Translate and solve:

    Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?

    Cevabı açıkla.

    $0.39

Symbols used here

\neq

not equal
The two sides are different.

\pm

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

\leq,\ \geq

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

\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 Equations using the Division and Multiplication Properties of Equality

1. Solve equations using the Division and Multiplication Properties of Equality
2. Solve equations that require simplification
3. Translate to an equation and solve
4. Translate and solve applications

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.

Kendini dene.

⌨ Type ✎ Draw

Matematik sorunun.

Çöz

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

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

Daha fazlası Algebra

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