1. Solving Linear Inequalities
5-5
Solving Linear Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Warm Up Graph each inequality. 1. x > –5 2. y ≤ 0
3. Write –6x + 2y = –4
in slope-intercept form,
and graph.

3. Objective
Graph and solve linear inequalities in two variables.

4. VOCABULARY
Linear inequality: similar to a linear equation, but the equal sign is replaced with an inequality symbol.
Solution of a linear inequality: any ordered pair that makes the inequality true.

5. Example 1A: Identifying Solutions of Inequalities
Tell whether the ordered pair is a solution of the inequality.
(–2, 4); y < 2x + 1
y < 2x + 1
4 2(–2) + 1
4 –4 + 1
4 –3
< 
Substitute (–2, 4) for (x, y).
(–2, 4) is not a solution.

6. Example 1B: Identifying Solutions of Inequalities
Tell whether the ordered pair is a solution of the inequality.
(3, 1); y > x – 4
y > x − 4
– 4
1 – 1
>
Substitute (3, 1) for (x, y). 
(3, 1) is a solution.

7. You try!!!!
Tell whether the ordered pair is a solution of the inequality.
a. (4, 5); y < x + 1
b. (1, 1); y > x – 7

8. When the inequality is written as:
the points on the boundary line _________________
of the inequality and the line is _____________
When the inequality is written as:
the points on the boundary line _________________
of the inequality and the line is _____________
STEP 1 – HOW SHOULD I DRAW THE BOUNDARY LINE?
𝒚≤𝒐𝒓 𝒚≥
𝒚<𝒐𝒓 𝒚>
are solutions
are not solutions
solid
dashed
When the inequality is written as:
the points _______ the boundary line are _________________
When the inequality is written as:
the points _______ the boundary line are _________________
𝒚>𝒐𝒓 𝒚≥
𝒚<𝒐𝒓 𝒚≤
STEP 2 – HOW SHOULD I SHADE?
above
below
solutions of the inequality
solutions of the inequality

9. Graphing Linear Inequalities
Step 1
Solve the inequality for y (slope-intercept form).
Step 2
Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >.
Step 3
Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.

10. Example 2A: Graphing Linear Inequalities in Two Variables
Graph the solutions of the linear inequality.
y  2x – 3
Step 1 The inequality is already solved for y.
Step 2 Graph the boundary line y = 2x – 3. Use a solid line for .
Step 3 The inequality is , so shade below the line.

11.  Example 2A Continued Graph the solutions of the linear inequality.
y  2x – 3
Substitute (0, 0) for (x, y) because it is not on the boundary line.
Check y  2x – 3
(0) – 3
0 –3  
A false statement means that the half-plane containing (0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

12. The point (0, 0) is a good test point to use if it does not lie on the boundary line.
Helpful Hint

13. Example 2B: Graphing Linear Inequalities in Two Variables
Graph the solutions of the linear inequality.
5x + 2y > –8
Step 1 Solve the inequality for y.
5x + 2y > –8
–5x –5x
2y > –5x – 8
y > x – 4
Step 2 Graph the boundary line Use a dashed line for >.
y = x – 4.

14. Example 2B Continued
Graph the solutions of the linear inequality.
5x + 2y > –8
Step 3 The inequality is >, so shade above the line.

15.  Example 2B Continued Graph the solutions of the linear inequality.
5x + 2y > –8
Substitute ( 0, 0) for (x, y) because it is not on the boundary line.
Check
y > x – 4
(0) – 4
0 –4
> 
The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

16. Try on your own!!!
Graph the solutions of the linear inequality.
2x – y – 4 > 0

17. Example 3: Application
Ada has at most 285 beads to make jewelry. A necklace requires 40 beads, and a bracelet requires 15 beads.
Write a linear inequality to describe the situation.
Let x represent the number of necklaces and y the number of bracelets.
Write an inequality. Use ≤ for “at most.”

18. Necklace beads bracelet
Example 3a Continued
Necklace
beads
bracelet
plus
is at
most
285
beads.
40x +
15y ≤
Solve the inequality for y.
40x + 15y ≤ 285
–40x –40x
15y ≤ –40x + 285
Subtract 40x from both sides.
Divide both sides by 15.

19. Example 3b
b. Graph the solutions. =
Step 1 Since Ada cannot make a negative amount of jewelry, the system is graphed only in Quadrant I. Graph the boundary line Use a solid line for ≤.

20. Example 3b Continued
b. Graph the solutions.
Step 2 Shade below the line. Ada can only make whole numbers of jewelry. All points on or below the line with whole number coordinates are the different combinations of bracelets and necklaces that Ada can make.

21. Example 3c
c. Give two combinations of necklaces and bracelets that Ada could make.
Two different combinations of jewelry that Ada could make with 285 beads could be 2 necklaces and 8 bracelets or 5 necklaces and 3 bracelets.
(2, 8)
(5, 3) 

22. Example Together
Write an inequality to represent the graph.
y-intercept: 1; slope:
Write an equation in slope-intercept form.
The graph is shaded above a dashed boundary line.
Replace = with > to write the inequality

23. Try on your own!!!
Write an inequality to represent the graph.
y-intercept: slope:

24. HOMEWORK PG
#12-21,
30-40(evens), 41, 42