Graphing Inequalities Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Graphing Inequalities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The process of representing the solution set of an inequality on a coordinate plane by drawing the boundary line (solid for $\leq$ / $\geq$ , dashed for $<$ / $>$ ) and shading the half-plane that contains all points satisfying the inequality.

Use boundary lines and shading to show where conditions are true.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Graphing an inequality means drawing its boundary line (solid if the endpoint counts, dashed if not) and shading the half-plane where the inequality holds.

Common stuck point: The procedure for graphing inequalities is the easy part; the trap is using a solid line for a strict inequality. Asking "Is the solution of a two-variable inequality a shaded region of the plane bounded by a line?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the solution of a two-variable inequality a shaded region of the plane bounded by a line?

Worked Examples

Example 1

easy

Graph

x > 2

on a number line.

Open circle at 2 (strict inequality), shaded ray to the right

Answer

Open circle at 2, shaded right →

(2, \infty)

First step

Place an open circle at

x = 2

(not included, strict inequality).

Full solution

2
Shade everything to the right of 2.
3
The arrow extends to $+\infty$ .

On a number line, open circles represent strict inequalities (

<, >

) and closed circles represent inclusive inequalities (

\leq, \geq

). Shade in the direction of the solution.

Example 2

medium

Graph the solution of

y \leq 2x + 1

on the coordinate plane.

Solid line y = 2x + 1; shade the region below (containing the origin)

Example 3

medium

Graph

y > \tfrac{1}{2}x - 1

on the coordinate plane.

Example 4

medium

Find the inequality whose graph is a dashed line through

(0,2)

and

(1,5)

, shaded below.

Example 5

medium

Graph

y \geq -x + 3

using a test point.

Example 6

medium

Graph the system

\{y < 2x + 1,\ y \geq -x - 2\}

Example 7

hard

Write the inequality whose graph is a solid line through

(-1,0)

and

(0,2)

, shaded above and to the left.

Example 8

hard

Graph

y \leq |x| - 1

on the coordinate plane.

Example 9

hard

Write a system of inequalities whose solution is the triangle with vertices

(0,0)

(4,0)

, and

(0,3)

Example 10

hard

A produce stand makes $2 per pound on apples and $3 per pound on pears. The owner can ship at most

50

pounds total and must ship at least

10

pounds of apples. Write the system of inequalities.

Example 11

challenge

Find the area of the region defined by

|x| + |y| \leq 2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Graph

x \leq -1

on a number line.

Closed circle at −1 (inclusive inequality), shaded ray to the left

Example 2

medium

Is the point

(1, 3)

in the solution region of

y < x + 4

Example 3

easy

When graphing

y<2x+1

, is the boundary line solid or dashed?

Boundary line y

Example 4

easy

When graphing

y\ge x-3

, is the boundary line solid or dashed?

Boundary line y

Example 5

easy

To graph

y>x

, do you shade above or below the line? (Test

(0,1)

Test point (0,1) is above y = x and satisfies y > x, so shade above the line

Example 6

easy

For

y\le-x+4

, does the point

(0,0)

satisfy it?

Origin (0,0) satisfies 0 ≤ −0+4 = 4; shade the side containing the origin

Example 7

easy

(2,5)

a solution of

y>2x

Point (2,5) lies above y = 2x since 5 > 4; it is in the solution region

Example 8

easy

What kind of line do you draw for

x\ge0

on the coordinate plane?

Solid vertical boundary at x = 0 (the y-axis); shade the right half-plane

Example 9

easy

For

y<3

, do you shade above or below the horizontal line

y=3

Dashed horizontal line y = 3; shade the region below (y < 3)

Example 10

easy

Is the line dashed or solid for

2x+3y>6

Example 11

medium

Graph

y\ge2x-1

: describe the line style and which side to shade (test

(0,0)

Solid line y = 2x − 1; origin satisfies 0 ≥ −1, so shade above (origin side)

Example 12

medium

For

y<-\frac{1}{2}x+2

, state the line style and shading side using

(0,0)

Dashed line y = −½x + 2; origin (0,0) satisfies 0 < 2, so shade below (origin side)

Example 13

medium

Where is the solution region of the system

y\ge0

and

x\ge0

Solid x-axis (y ≥ 0) and y-axis (x ≥ 0); solution region is the first quadrant including axes

Example 14

medium

Test

(1,1)

for the system

y<x+2

and

y>x-1

. Is it in the solution region?

Point (1,1) is between the two parallel lines and satisfies both y < x+2 and y > x−1

Example 15

medium

For

3x-2y\le6

, rewrite in

y

-form and state the shading side (test

(0,0)

Solid line y = (3/2)x − 3; origin satisfies 0 ≥ −3, so shade above

Example 16

medium

Does the boundary of

y\le2x

pass through the origin, and is the origin a usable test point?

Example 17

medium

Graph

|x|\le2

on the coordinate plane — describe the region.

Solid vertical boundaries at x = −2 and x = 2; solution is the strip −2 ≤ x ≤ 2

Example 18

medium

For the system

y\le4

and

y\ge1

, describe the solution region.

Solid horizontal lines at y = 1 and y = 4; solution is the band 1 ≤ y ≤ 4

Example 19

medium

For

2x+y\le4

, rewrite in

y

-form and state the shading side (test

(0,0)

Solid line y = −2x + 4; origin satisfies 0 ≤ 4, so shade below (origin side)

Example 20

challenge

A factory needs

x\ge0

y\ge0

, and

x+y\le8

. Identify the vertices of the feasible region.

Feasible region bounded by x ≥ 0, y ≥ 0, and x+y ≤ 8; vertices at (0,0), (8,0), (0,8)

Example 21

challenge

Without graphing, determine whether the systems

y>x

and

y<x-2

have any common solution.

Example 22

challenge

For the system

y\ge x^2

and

y\le4

, describe the solution region.

Example 23

easy

Graph

x \geq 4

on a number line.

Example 24

easy

(0,0)

in the solution region of

y \geq x - 1

Example 25

easy

Graph

y > -2

on the coordinate plane.

Example 26

easy

(2,1)

a solution of

y \leq 3x - 4

Example 27

medium

Which inequality has the point

(3, 2)

on its solid boundary?

Example 28

medium

Graph

2x - y \geq 4

. Where is the shaded region relative to the boundary?

Example 29

medium

(2, 3)

in the solution region of

3x + 2y \leq 5

Example 30

medium

Find the slope and

y

-intercept of the boundary of

4x + 2y < 8

Example 31

medium

On a number line, graph

-2 < x \leq 3

Example 32

hard

For the system

\{x \geq 0,\ y \geq 0,\ x + y \leq 4\}

, what is the area of the solution region?

Example 33

hard

For which value of

k

does the line

y = 2x + k

pass through

(3, 4)

as the solid boundary of an inequality containing

(0,0)

Example 34

hard

How many integer points

(x, y)

satisfy

x \geq 0

y \geq 0

, and

x + y \leq 3

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Related Concepts

Inequalities Coordinate Plane Number Line

Background Knowledge

These ideas may be useful before you work through the harder examples.

inequalitiescoordinate planenumber line