Graphing Inequalities in Two Variables 12-6 Learn to graph inequalities on the coordinate plane.

Graphing Inequalities in Two Variables12-6

Learn to graph inequalities on the coordinate plane.


Graph each inequality.

y < x – 1

Example 1: Graphing Inequalities

First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed. Then determine on which side of the line the solutions lie.

(0, 0)

y < x – 1

Test a point not on the line.

Substitute 0 for x and 0 for y.0 < 0 – 1?

0 < –1?


Any point on the line y = x 1 is not a solution of y < x 1 because the inequality symbol < means only “less than” and does not include “equal to.”

Helpful Hint


Example 1 Continued

(0, 0)

Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0).


y 2x + 1


First graph the boundary line y = 2x + 1. Since points that are on the line are solutions of y 2x + 1, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 2x + 1 lie.

(0, 4) Choose any point not on the line.

Substitute 0 for x and 4 for y.

y ≥ 2x + 1

4 ≥ 0 + 1?


Any point on the line y = 2x 1 is a solution of y ≥ 2x 1 because the inequality symbol ≥ means “greater than or equal to.”

Helpful Hint


Example 2 Continued

Since 4 1 is true, (0, 4) is a solution of y 2x + 1. Shade the side of the line that includes (0, 4).

(0, 4)


2y + 5x < 6


First write the equation in slope-intercept form.

2y < –5x + 6

2y + 5x < 6

y < – x + 352

Then graph the line y = – x + 3. Since points that

are on the line are not solutions of y < – x + 3,

make the line dashed. Then determine on which

side of the line the solutions lie.

52 5

2

Subtract 5x from both sides.

Divide both sides by 2.


Example 3 Continued

Since 0 < 3 is true, (0, 0) is a

solution of y < – x + 3.

Shade the side of the line

that includes (0, 0).

52


y < – x + 352

0 < 0 + 3?

0 < 3?


(0, 0)



y < x – 4

Example 4

First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie.

(0, 0)

y < x – 4

Test a point not on the line.

Substitute 0 for x and 0 for y.0 < 0 – 4?

0 < –4?


Example 4 Continued

(0, 0)

Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).


y > 4x + 4

Example 5

First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of y 4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 4x + 4 lie.



y ≥ 4x + 4

3 ≥ 8 + 4?


Example 5 Continued

Since 3 12 is not true, (2, 3) is not a solution of y 4x + 4. Shade the side of the line that does not include (2, 3).

(2, 3)


3y + 4x 9

Example 6

First write the equation in slope-intercept form.

3y –4x + 9

3y + 4x 9

y – x + 343

Subtract 4x from both sides.

Divide both sides by 3.

43Then graph the line y = – x + 3. Since points that

are on the line are solutions of y – x + 3, make

the line solid. Then determine on which side of the

line the solutions lie.

43


Example 6 Continued

Since 0 3 is not true, (0, 0) is

not a solution of y – x + 3.

Shade the side of the line that

does not include (0, 0).

43


y – x + 343

0 0 + 3?

0 3?


(0, 0)


Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 a week for spending money. How many weeks can Keith withdraw money in his account and still have at least $200 in at the end of summer?

Example 7: Real World


The phrase “no more” can be translated as less than or equal to.

Helpful Hint


Example 8: Real World

A taxi charges a $1.75 flat rate fee in addition to $0.65 per mile. Katie has no more than $15 to spend. How many miles can Katie travel without going over what she has to spend?


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems



1. y < – x + 4

13

Lesson Quiz Part I


2. 4y + 2x > 12

Lesson Quiz Part II


1. Identify the graph of the given inequality. 6y + 3x > 12

A. B.



2. Tell which ordered pair is a solution of the inequality y < x + 12.

A. (–3, 5)

B. (–4, 12)

C. (–5, 8)

D. (–7, 9)


Graphing Inequalities in Two Variables 12-6 Learn to graph inequalities on the coordinate plane.

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