1. 7.4 Linear Inequalities in Two Variables
Page 233

2. Vocabulary
Linear inequality – an inequality that can be written with one of the inequality signs, <, >, ≤, ≥.
Solution of an inequality – any ordered pair (x, y) that makes the inequality true.
Boundary line – a line that divides a coordinate plane into two half planes.
Half-plane – the part of the coordinate plane on one side of a line, which may include the line.

3. Tips Symbol Type of Line Where to Shade < Dashed line
Shade below the line
>
Shade above the line ≤
Solid line ≥

4. Example 1
Graph the solution set for 6x – 3y ≤ 12. Step 1: Rewrite the equation in slope-intercept form (y = mx + b). 6x – 3y ≤ 12 -6x -6x  Subtract 6x from both sides -3y ≤ -6x + 12 −3𝑦 −3 ≤ −6𝑥 − −3  Divide by -3.

5. −3𝑦 −3 ≤ −6𝑥 − −3  Divide by (Reverse the inequality sign when you divide by a negative number.)
y ≥ 2x – 4
Step 2: Graph the boundary line.
The slope is 2 and the y-intercept is -4. Use this information to graph the two points needed to draw your line.
y ≥ 2x – 4 uses the inequality ≥, so the line should be solid. Therefore, draw a solid line through the two points.
Step 3: Shade the appropriate area.
y ≥ 2x – 4 uses the inequality ≥, so shade above the solid line.

6. To check your work, choose one point above the boundary line
To check your work, choose one point above the boundary line. Substitute both points into the original inequality, and verify that the point above the boundary line makes the inequality true.
Please select a point so that we can check if it’s a solution of the inequality.

7. Your Turn
Now it’s your turn. Do Your Turn #2 on page 234.

8. Your Turn Answers

9. Example 2
Graph the solution set for x + 5y < 30. Step 1: Rewrite the equation in slope-intercept form (y = mx + b). x + 5y < 30 - x - x  Subtract x from both sides. 5y < -x 𝑦 5 < −1𝑥 Divide by 5.

10. 5𝑦 5 < −1𝑥 Divide by 5.
y < - 𝟏 𝟓 x + 6
Step 2: Graph the boundary line.
The slope is - 𝟏 𝟓 and the y-intercept is 6. Use this information to graph the two points needed to draw your line.
y < - 𝟏 𝟓 x + 6 uses the inequality <, so use a dashed line to show that the points on the line are not a part of the solution.
Step 3: Shade the appropriate area.
y < - 𝟏 𝟓 x + 6 uses the inequality <, so shade below the dashed line.

11. To check your work, choose one point above the boundary line
To check your work, choose one point above the boundary line. Substitute both points into the original inequality, and verify that the point above the boundary line makes the inequality true.
Please select a point so that we can check if it’s a solution of the inequality.

12. Your Turn
Now it’s your turn. Do Your Turn #5 on page 235.

13. Your Turn Answers

14. Writing and Solving Linear Inequalities
Kyra’s variety show can last up to 44 minutes and has 4 minutes of intro and credits combined. The show features y comedy segments that last 8 minutes each or x music segments that last 3 minutes each. Write an inequality to represent the situation. Then graph the inequality.

15. Kyra’s variety show can last up to 44 minutes (≤ 44) and has 4 minutes of intro and credits combined (+ 4). The show features y comedy segments that last 8 minutes each (8y) or x music segments that last 3 minutes each (3x). Write an inequality to represent the situation. Then graph the inequality.
Step 1: Write a linear inequality to describe the situation.
3x + 8y + 4 ≤ 44
Solve the inequality for y.
 Subtract 4 from both sides.
3x + 8y ≤ 40

16. 3x + 8y ≤
-3x x  Subtract 3x from both sides.
8y ≤ -3x + 40
8 8 y ≤ −3𝑥  Divide by 8.
y ≤ - 𝟑 𝟖 x + 5
Step 2: Graph the boundary line.
The slope is - 𝟑 𝟖 and the y-intercept is 5. Use this information to graph the two points needed to draw your line.
y < - 𝟑 𝟖 x + 5 uses the inequality ≤, so use a solid line to show that the points on the line are not a part of the solution.

17. Step 3: Shade the appropriate area.
y ≤ - 𝟑 𝟖 x + 5 uses the inequality ≤, so shade below the solid line.

18. Your Turn
Now it’s your turn. Do Your Turn #7-9 on page 237.

19. Your Turn Answers