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Graph Linear Inequalities in Two Variables

Verify solutions to an inequality in two variables.

Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality $x>3$ is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See .

Similarly, linear inequalities in two variables have many solutions. Any ordered pair $(x,y)$ that makes an inequality true when we substitute in the values is a solution to a linear inequality.

Example

Try it.

Determine whether each ordered pair is a solution to the inequality $y>x+4:$

ⓐ $(0,0)$ ⓑ $(1,6)$ ⓒ $(2,6)$ ⓓ $(-5,-15)$ ⓔ $(-8,12)$

Solution

ⓐ

$(0,0)$
$\$
Simplify.
	So, $(0,0)$ is not a solution to $y>x+4.$

ⓑ

$(1,6)$
$\$
Simplify.
	So, $(1,6)$ is a solution to $y>x+4.$

ⓒ

$(2,6)$
$\$
Simplify.
	So, $(2,6)$ is not a solution to $y>x+4.$

ⓓ

$(-5,-15)$

Simplify.
	So, $(-5,-15)$ is not a solution to $y>x+4.$

ⓔ

$(-8,12)$
$\$
Simplify.
	So, $(-8,12)$ is a solution to $y>x+4.$

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in shown previously again. The point $x=3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See .

Similarly, the line $y=x+4$ separates the plane into two regions. On one side of the line are points with $yx+4.$ We call the line $y=x+4$ a boundary line.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

\[\begin{array}{llllll}Ax+ByC & & & & & Ax+By\ge C \\ \text{Boundary line is}\ Ax+By=C & & & & & \text{Boundary line is}\ Ax+By=C \\ \text{Boundary line is not included in solution.} & & & & & \text{Boundary line is included in solution.} \\ \text{Boundary line is dashed.} & & & & & \text{Boundary line is solid.}\end{array}\]

Now, let’s take a look at what we found in . We’ll start by graphing the line $y=x+4,$ and then we’ll plot the five points we tested, as shown in the graph. See .

In we found that some of the points were solutions to the inequality $y>x+4$ and some were not.

\[\begin{array}{lll}y & > & x+4 \\ 10 & \overset{?}{>} & 0+4 \\ 10 & > & 4\end{array}\]

Example

Try it.

The boundary line shown in this graph is $y=2x-1.$ Write the inequality shown by the graph.

Solution

The line $y=2x-1$ is the boundary line. On one side of the line are the points with $y>2x-1$ and on the other side of the line are the points with $y<2x-1.$

Let’s test the point $(0,0)$ and see which inequality describes its position relative to the boundary line.

At $(0,0),$ which inequality is true: $y>2x-1$ or $y<2x-1?$

\[\begin{array}{llllll}y>2x-1 & & & & & y<2x-1 \\ 0\overset{?}{>}2\cdot 0-1 & & & & & 0\overset{?}{<}2\cdot 0-1 \\ 0>-1\ \text{True} & & & & & 0<-1\ \text{False}\end{array}\]

Since, $y>2x-1$ is true, the side of the line with $(0,0),$ is the solution. The shaded region shows the solution of the inequality $y>2x-1.$

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality $y\ge 2x-1.$

We could use any point as a test point, provided it is not on the line. Why did we choose $(0,0)?$ Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that $y<2x-1.$

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

Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

How to Graph a Linear Equation in Two Variables

Try it.

Graph the linear inequality $y\ge \frac{3}{4}x-2.$

Solution

The steps we take to graph a linear inequality are summarized here.

Example

Try it.

Graph the linear inequality $x-2y<5.$

Solution

First, we graph the boundary line $x-2y=5.$ The inequality is $<$ so we draw a dashed line.

Then, we test a point. We’ll use $(0,0)$ again because it is easy to evaluate and it is not on the boundary line.

Is $(0,0)$ a solution of $x-2y<5?$

The point $(0,0)$ is a solution of $x-2y<5,$ so we shade in that side of the boundary line.

All points in the shaded region, but not those on the boundary line, represent the solutions to $x-2y<5.$

What if the boundary line goes through the origin? Then, we won’t be able to use $(0,0)$ as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Example

Try it.

Graph the linear inequality: $y\le \text{}-4x.$

Solution

First, we graph the boundary line $y=-4x.$ It is in slope–intercept form, with $m=-4$ and $b=0.$ The inequality is $\le$ so we draw a solid line.

Now we need a test point. We can see that the point $(1,0)$ is not on the boundary line.

Is $(1,0)$ a solution of $y\le -4x?$

The point $(1,0)$ is not a solution to $y\le \text{}-4x,$ so we shade in the opposite side of the boundary line.

All points in the shaded region and on the boundary line represent the solutions to $y\le \text{}-4x.$

Recall that:

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

Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Example

Try it.

Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least $240 a week. Her job in food service pays $10 an hour and her tutoring job on campus pays $15 an hour. How many hours does Hilaria need to work at each job to earn at least $240?

ⓐ Let $x$ be the number of hours she works at the job in food service and let y be the number of hours she works tutoring. Write an inequality that would model this situation.

ⓑ Graph the inequality.

Solution

ⓐ We let x be the number of hours she works at the job in food service and let y be the number of hours she works tutoring.

She earns $10 per hour at the job in food service and $15 an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.

ⓑ To graph the inequality, we put it in slope–intercept form.

\[\begin{array}{lll}10x+15y & \ge & 240 \\ 15y & \ge & -10x+240 \\ y & \ge & -\frac{2}{3}x+16\end{array}\]

For Hilaria, it means that to earn at least $240, she can work 15 hours tutoring and 10 hours at her fast-food job, earn all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service.

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

Key Concepts

How to graph a linear inequality in two variables.
1. Identify and graph the boundary line.
  If the inequality is $\le \text{or}\ge ,$ the boundary line is solid.
  If the inequality is $<\ \text{or}>,$ the boundary line is dashed.
2. Test a point that is not on the boundary line. Is it a solution of the inequality?
3. Shade in one side of the boundary line.
  If the test point is a solution, shade in the side that includes the point.
  If the test point is not a solution, shade in the opposite side.

Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Try it.

Determine whether each ordered pair is a solution to the inequality $y>x-1:$

ⓐ $(0,1)$
ⓑ $(-4,-1)$
ⓒ $(4,2)$
ⓓ $(3,0)$
ⓔ $(-2,-3)$

Solution

ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ no

Try it.

Determine whether each ordered pair is a solution to the inequality $y>x-3:$

ⓐ $(0,0)$
ⓑ $(2,1)$
ⓒ $(-1,-5)$
ⓓ $(-6,-3)$
ⓔ $(1,0)$

Try it.

Determine whether each ordered pair is a solution to the inequality $y<3x+2:$

ⓐ $(0,3)$
ⓑ $(-3,-2)$
ⓒ $(-2,0)$
ⓓ $(0,0)$
ⓔ $(-1,4)$

Solution

ⓐ no ⓑ no ⓒ no ⓓ yes ⓔ no

Try it.

Determine whether each ordered pair is a solution to the inequality $y<-2x+5:$

ⓐ $(-3,0)$
ⓑ $(1,6)$
ⓒ $(-6,-2)$
ⓓ $(0,1)$
ⓔ $(5,-4)$

Try it.

Determine whether each ordered pair is a solution to the inequality $3x-4y>4:$

ⓐ $(5,1)$
ⓑ $(-2,6)$
ⓒ $(3,2)$
ⓓ $(10,-5)$
ⓔ $(0,0)$

Solution

ⓐ yes ⓑ no ⓒ no ⓓ yes ⓔ no

Try it.

Determine whether each ordered pair is a solution to the inequality $2x+3y>2:$

ⓐ $(1,1)$
ⓑ $(4,-3)$
ⓒ $(0,0)$
ⓓ $(-8,12)$
ⓔ $(3,0)$

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Try it.

Write the inequality shown by the graph with the boundary line $y=3x-4.$

Solution

$y\le 3x-4$

Try it.

Write the inequality shown by the graph with the boundary line $y=2x-4.$

Try it.

Write the inequality shown by the graph with the boundary line $y=\frac{1}{2}x+1.$

Solution

$y\le \frac{1}{2}x+1$

Try it.

Write the inequality shown by the graph with the boundary line $y=-\frac{1}{3}x-2.$

Try it.

Write the inequality shown by the shaded region in the graph with the boundary line $x+y=5.$

Solution

$x+y\ge 5$

Try it.

Write the inequality shown by the shaded region in the graph with the boundary line $x+y=3.$

Try it.

Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6.$

Solution

$3x-y\le 6$

Try it.

Write the inequality shown by the shaded region in the graph with the boundary line $2x-y=4.$

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

Try it.

Graph the linear inequality: $y>\frac{2}{3}x-1.$

Solution

Try it.

Graph the linear inequality: $y<\frac{3}{5}x+2.$

Try it.

Graph the linear inequality: $y\le -\frac{1}{2}x+4.$

Solution

Try it.

Graph the linear inequality: $y\ge -\frac{1}{3}x-2.$

Try it.

Graph the linear inequality: $x-y\le 3.$

Solution

Try it.

Graph the linear inequality: $x-y\ge -2.$

Try it.

Graph the linear inequality: $4x+y>-4.$

Solution

Try it.

Graph the linear inequality: $x+5y<-5.$

Try it.

Graph the linear inequality: $3x+2y\ge -6.$

Solution

Try it.

Graph the linear inequality: $4x+2y\ge -8.$

Try it.

Graph the linear inequality: $y>4x.$

Solution

Try it.

Graph the linear inequality: $y\le -3x.$

Try it.

Graph the linear inequality: $y<-10.$

Solution

Try it.

Graph the linear inequality: $y\ge 2.$

Try it.

Graph the linear inequality: $x\le 5.$

Solution

Try it.

Graph the linear inequality: $x\ge 0.$

Try it.

Graph the linear inequality: $x-y<4.$

Solution

Try it.

Graph the linear inequality: $x-y<-3.$

Try it.

Graph the linear inequality: $y\ge \frac{3}{2}x.$

Solution

Try it.

Graph the linear inequality: $y\le \frac{5}{4}x.$

Try it.

Graph the linear inequality: $y>-2x+1.$

Solution

Try it.

Graph the linear inequality: $y<-3x-4.$

Try it.

Graph the linear inequality: $2x+y\ge -4.$

Solution

Try it.

Graph the linear inequality: $x+2y\le -2.$

Try it.

Graph the linear inequality: $2x-5y>10.$

Solution

Try it.

Graph the linear inequality: $4x-3y>12.$

Solve Applications using Linear Inequalities in Two Variables

Try it.

Harrison works two part time jobs. One at a gas station that pays $11 an hour and the other is IT troubleshooting for $\text{\$}16.50$ an hour. Between the two jobs, Harrison wants to earn at least $330 a week. How many hours does Harrison need to work at each job to earn at least $330?

ⓐ Let x be the number of hours he works at the gas station and let y be the number of (hours he works troubleshooting. Write an inequality that would model this situation.

ⓑ Graph the inequality.

Solution

Try it.

Elena needs to earn at least $450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays $9 an hour and the other as an intern in a genetics lab for $22.50 per hour. How many hours does Elena need to work at each job to earn at least $450 per week?

ⓐ Let x be the number of hours she works teaching swimming and let y be the number of hours she works as an intern. Write an inequality that would model this situation.

ⓑ Graph the inequality.

Try it.

The doctor tells Laura she needs to exercise enough to burn 500 calories each day. She prefers to either run or bike and burns 15 calories per minute while running and 10 calories a minute while biking.

ⓐ If x is the number of minutes that Laura runs and y is the number minutes she bikes, find the inequality that models the situation.

ⓑ Graph the inequality.

Solution

Try it.

Armando’s workouts consist of kickboxing and swimming. While kickboxing, he burns 10 calories per minute and he burns 7 calories a minute while swimming. He wants to burn 600 calories each day.

ⓐ If x is the number of minutes that Armando will kickbox and y is the number minutes he will swim, find the inequality that will help Armando create a workout for today.

ⓑ Graph the inequality.

Condensed — the full section is in OpenStax Intermediate 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.

Graph $x>2$ on a number line.
If you missed this problem, review .

Ipahayag ang sagot
Solve: $4x+3>23$.
If you missed this problem, review .

Ipahayag ang sagot

$x>5$
Translate: $8>x>3$.
If you missed this problem, review .

Ipahayag ang sagot

$x$ is between 3 and 8, not inclusive
Determine whether each ordered pair is a solution to the inequality $y>x+4:$
ⓐ $(0,0)$ ⓑ $(1,6)$ ⓒ $(2,6)$ ⓓ $(-5,-15)$ ⓔ $(-8,12)$

Ipahayag ang sagot

ⓐ

$(0,0)$
$\$
Simplify.
So, $(0,0)$ is not a solution to $y>x+4.$

ⓑ

$(1,6)$
$\$
Simplify.
So, $(1,6)$ is a solution to $y>x+4.$

ⓒ

$(2,6)$
$\$
Simplify.
So, $(2,6)$ is not a solution to $y>x+4.$

ⓓ

$(-5,-15)$
Simplify.
So, $(-5,-15)$ is not a solution to $y>x+4.$

ⓔ

$(-8,12)$
$\$
Simplify.
So, $(-8,12)$ is a solution to $y>x+4.$
Determine whether each ordered pair is a solution to the inequality $y>x-3:$
ⓐ $(0,0)$ ⓑ $(4,9)$ ⓒ $(-2,1)$ ⓓ $(-5,-3)$ ⓔ $(5,1)$

Ipahayag ang sagot

ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no
Determine whether each ordered pair is a solution to the inequality $y
ⓐ \((0,0)$ ⓑ $(8,6)$ ⓒ $(-2,-1)$ ⓓ $(3,4)$ ⓔ $(-1,-4)$

Ipahayag ang sagot

ⓐ yes ⓑ yes ⓒ no ⓓ no
ⓔ yes
The boundary line shown in this graph is $y=2x-1.$ Write the inequality shown by the graph.

Ipahayag ang sagot

The line $y=2x-1$ is the boundary line. On one side of the line are the points with $y>2x-1$ and on the other side of the line are the points with $y<2x-1.$
Let’s test the point $(0,0)$ and see which inequality describes its position relative to the boundary line.
At $(0,0),$ which inequality is true: $y>2x-1$ or $y<2x-1?$
\[\begin{array}{llllll}y>2x-1 & & & & & y<2x-1 \\ 0\overset{?}{>}2\cdot 0-1 & & & & & 0\overset{?}{<}2\cdot 0-1 \\ 0>-1\ \text{True} & & & & & 0<-1\ \text{False}\end{array}\]
Since, $y>2x-1$ is true, the side of the line with $(0,0),$ is the solution. The shaded region shows the solution of the inequality $y>2x-1.$
Since the boundary line is graphed with a solid line, the inequality includes the equal sign.
The graph shows the inequality $y\ge 2x-1.$
We could use any point as a test point, provided it is not on the line. Why did we choose $(0,0)?$ Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that $y<2x-1.$
Write the inequality shown by the graph with the boundary line $y=-2x+3.$

Ipahayag ang sagot

$y\ge -2x+3$
Write the inequality shown by the graph with the boundary line $y=\frac{1}{2}x-4.$

Ipahayag ang sagot

$y\le \frac{1}{2}x-4$
The boundary line shown in this graph is $2x+3y=6.$ Write the inequality shown by the graph.

Ipahayag ang sagot

The line $2x+3y=6$ is the boundary line. On one side of the line are the points with $2x+3y>6$ and on the other side of the line are the points with $2x+3y<6.$
Let’s test the point $(0,0)$ and see which inequality describes its side of the boundary line.
At $(0,0),$ which inequality is true: $2x+3y>6$ or $2x+3y<6?$
\[\begin{array}{llllllllllllllllll}\begin{array}{lll}2x+3y & > & 6 \\ 2(0)+3(0) & \overset{?}{>} & 6 \\ 0 & > & 6\ \text{False}\end{array} & & & & & \begin{array}{lll}2x+3y & < & 6 \\ 2(0)+3(0) & \overset{?}{<} & 6 \\ 0 & < & 6\ \text{True}\end{array}\end{array}\]
So the side with $(0,0)$ is the side where $2x+3y<6.$
(You may want to pick a point on the other side of the boundary line and check that $2x+3y>6.$)
Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.
The shaded region shows the solution to the inequality $2x+3y<6.$
Write the inequality shown by the shaded region in the graph with the boundary line $x-4y=8.$

Ipahayag ang sagot

$x-4y\le 8$
Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6.$

Ipahayag ang sagot

$3x-y\ge 6$
Graph the linear inequality $y\ge \frac{3}{4}x-2.$
Graph the linear inequality $y\ge \frac{5}{2}x-4.$

Ipahayag ang sagot

All points in the shaded region and on the boundary line, represent the solutions to $y>\frac{5}{2}x-4.$
Graph the linear inequality $y<\frac{2}{3}x-5.$

Ipahayag ang sagot

All points in the shaded region, but not those on the boundary line, represent the solutions to $y<\frac{2}{3}x-5.$
Graph the linear inequality $x-2y<5.$

Ipahayag ang sagot

First, we graph the boundary line $x-2y=5.$ The inequality is $<$ so we draw a dashed line.

Then, we test a point. We’ll use $(0,0)$ again because it is easy to evaluate and it is not on the boundary line.
Is $(0,0)$ a solution of $x-2y<5?$
The point $(0,0)$ is a solution of $x-2y<5,$ so we shade in that side of the boundary line.

All points in the shaded region, but not those on the boundary line, represent the solutions to $x-2y<5.$
Graph the linear inequality: $2x-3y<6.$

Ipahayag ang sagot

All points in the shaded region, but not those on the boundary line, represent the solutions to $2x-3y<6.$
Graph the linear inequality: $2x-y>3.$

Ipahayag ang sagot

All points in the shaded region, but not those on the boundary line, represent the solutions to $2x-y>3.$
Graph the linear inequality: $y\le \text{}-4x.$

Ipahayag ang sagot

First, we graph the boundary line $y=-4x.$ It is in slope–intercept form, with $m=-4$ and $b=0.$ The inequality is $\le$ so we draw a solid line.

Now we need a test point. We can see that the point $(1,0)$ is not on the boundary line.
Is $(1,0)$ a solution of $y\le -4x?$

The point $(1,0)$ is not a solution to $y\le \text{}-4x,$ so we shade in the opposite side of the boundary line.

All points in the shaded region and on the boundary line represent the solutions to $y\le \text{}-4x.$
Graph the linear inequality: $y>-3x.$

Ipahayag ang sagot

All points in the shaded region, but not those on the boundary line, represent the solutions to $y>-3x.$
Graph the linear inequality: $y\ge -2x.$

Ipahayag ang sagot

All points in the shaded region and on the boundary line, represent the solutions to $y\ge -2x.$
Graph the linear inequality: $y>3.$

Ipahayag ang sagot

First, we graph the boundary line $y=3.$ It is a horizontal line. The inequality is $>$ so we draw a dashed line.
We test the point $(0,0).$
\[\begin{array}{l} \\ y>3 \\ 0>3\end{array}\]
So, $(0,0)$ is not a solution to $y>3.$
So we shade the side that does not include $(0,0)$ as shown in this graph.

All points in the shaded region, but not those on the boundary line, represent the solutions to $y>3.$
Graph the linear inequality: $y<5.$

Ipahayag ang sagot

All points in the shaded region, but not those on the boundary line, represent the solutions to $y<5.$
Graph the linear inequality: $y\le -1.$

Ipahayag ang sagot

All points in the shaded region and on the boundary line represent the solutions to $y\le -1.$
Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least $240 a week. Her job in food service pays $10 an hour and her tutoring job on campus pays $15 an hour. How many hours does Hilaria need to work at each job to earn at least $240?
ⓐ Let $x$ be the number of hours she works at the job in food service and let y be the number of hours she works tutoring. Write an inequality that would model this situation.
ⓑ Graph the inequality.
ⓒ Find three ordered pairs $(x,y)$ that would be solutions to the inequality. Then, explain what that means for Hilaria.

Ipahayag ang sagot

ⓐ We let x be the number of hours she works at the job in food service and let y be the number of hours she works tutoring.
She earns $10 per hour at the job in food service and $15 an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.

ⓑ To graph the inequality, we put it in slope–intercept form.
\[\begin{array}{lll}10x+15y & \ge & 240 \\ 15y & \ge & -10x+240 \\ y & \ge & -\frac{2}{3}x+16\end{array}\]

ⓒ From the graph, we see that the ordered pairs $(15,10),(0,16),(24,0)$ represent three of infinitely many solutions. Check the values in the inequality.

For Hilaria, it means that to earn at least $240, she can work 15 hours tutoring and 10 hours at her fast-food job, earn all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service.
Hugh works two part time jobs. One at a grocery store that pays $10 an hour and the other is babysitting for $13 hour. Between the two jobs, Hugh wants to earn at least $260 a week. How many hours does Hugh need to work at each job to earn at least $260?
ⓐ Let x be the number of hours he works at the grocery store and let y be the number of hours he works babysitting. Write an inequality that would model this situation.
ⓑ Graph the inequality.
ⓒ Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Hugh.

Ipahayag ang sagot

ⓐ $10x+13y\ge 260$
ⓑ

ⓒ Answers will vary.
Veronica works two part time jobs in order to earn enough money to meet her obligations of at least $280 a week. Her job at the day spa pays $10 an hour and her administrative assistant job on campus pays $17.50 an hour. How many hours does Veronica need to work at each job to earn at least $280?
ⓐ Let x be the number of hours she works at the day spa and let y be the number of hours she works as administrative assistant. Write an inequality that would model this situation.
ⓑ Graph the inequality.
ⓒ Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Veronica

Ipahayag ang sagot

ⓐ $10x+17.5y\ge 280$
ⓑ

ⓒ Answers will vary.
Determine whether each ordered pair is a solution to the inequality $y>x-1:$

ⓐ $(0,1)$
ⓑ $(-4,-1)$
ⓒ $(4,2)$
ⓓ $(3,0)$
ⓔ $(-2,-3)$

Ipahayag ang sagot

ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ no
Determine whether each ordered pair is a solution to the inequality $y>x-3:$

ⓐ $(0,0)$
ⓑ $(2,1)$
ⓒ $(-1,-5)$
ⓓ $(-6,-3)$
ⓔ $(1,0)$
Determine whether each ordered pair is a solution to the inequality $y<3x+2:$

ⓐ $(0,3)$
ⓑ $(-3,-2)$
ⓒ $(-2,0)$
ⓓ $(0,0)$
ⓔ $(-1,4)$

Ipahayag ang sagot

ⓐ no ⓑ no ⓒ no ⓓ yes ⓔ no
Determine whether each ordered pair is a solution to the inequality $y<-2x+5:$

ⓐ $(-3,0)$
ⓑ $(1,6)$
ⓒ $(-6,-2)$
ⓓ $(0,1)$
ⓔ $(5,-4)$
Determine whether each ordered pair is a solution to the inequality $3x-4y>4:$

ⓐ $(5,1)$
ⓑ $(-2,6)$
ⓒ $(3,2)$
ⓓ $(10,-5)$
ⓔ $(0,0)$

Ipahayag ang sagot

ⓐ yes ⓑ no ⓒ no ⓓ yes ⓔ no
Determine whether each ordered pair is a solution to the inequality $2x+3y>2:$

ⓐ $(1,1)$
ⓑ $(4,-3)$
ⓒ $(0,0)$
ⓓ $(-8,12)$
ⓔ $(3,0)$
Write the inequality shown by the graph with the boundary line $y=3x-4.$

Ipahayag ang sagot

$y\le 3x-4$
Write the inequality shown by the graph with the boundary line $y=2x-4.$
Write the inequality shown by the graph with the boundary line $y=\frac{1}{2}x+1.$

Ipahayag ang sagot

$y\le \frac{1}{2}x+1$
Write the inequality shown by the graph with the boundary line $y=-\frac{1}{3}x-2.$
Write the inequality shown by the shaded region in the graph with the boundary line $x+y=5.$

Ipahayag ang sagot

$x+y\ge 5$
Write the inequality shown by the shaded region in the graph with the boundary line $x+y=3.$
Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6.$

Ipahayag ang sagot

$3x-y\le 6$

Symbols used here

\leq,\ \geq

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

\pm

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

\neq

not equal
The two sides are different.

\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.

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: Graph Linear Inequalities in Two Variables

Verify solutions to an inequality in two variables.
Recognize the relation between the solutions of an inequality and its graph.
Graph linear inequalities in two variables
Solve applications using linear inequalities in two variables
Identify and graph the boundary line.
If the inequality is
If the inequality is
Test a point that is not on the boundary line. Is it a solution of the inequality?

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.

Subukan ang iyong sarili

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

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\((0,0)\)
\(\\)
Simplify.
	So, \((0,0)\) is not a solution to \(y>x+4.\)

\((1,6)\)
\(\\)
Simplify.
	So, \((1,6)\) is a solution to \(y>x+4.\)

\((2,6)\)
\(\\)
Simplify.
	So, \((2,6)\) is not a solution to \(y>x+4.\)

\((-8,12)\)
\(\\)
Simplify.
	So, \((-8,12)\) is a solution to \(y>x+4.\)

\((0,0)\)
\(\\)
Simplify.
	So, \((0,0)\) is not a solution to \(y>x+4.\)

\((-5,-15)\)

Simplify.
	So, \((-5,-15)\) is not a solution to \(y>x+4.\)