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

Holt McDougal Algebra 1

5-5 Solving Linear Inequalities5-5 Solving Linear Inequalities

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz



5-5 Solving Linear Inequalities

Warm UpGraph each inequality.1. x > –5 2. y ≤ 0

3. Write –6x + 2y = –4 in slope-intercept form, and graph.



Graph and solve linear inequalities in two variables.

Objective



VOCABULARY

1. Linear inequality: similar to a linear equation, but the equal sign is replaced with an inequality symbol.

2. Solution of a linear inequality: any ordered pair that makes the inequality true.



Tell whether the ordered pair is a solution of the inequality.

Example 1A: Identifying Solutions of Inequalities

(–2, 4); y < 2x + 1

Substitute (–2, 4) for (x, y).

y < 2x + 1

4 2(–2) + 1

4 –4 + 14 –3<

(–2, 4) is not a solution.




Example 1B: Identifying Solutions of Inequalities

(3, 1); y > x – 4

Substitute (3, 1) for (x, y).

y > x − 4

1 3 – 4

1 – 1>

(3, 1) is a solution.


5-5 Solving Linear InequalitiesYou try!!!!

a. (4, 5); y < x + 1


b. (1, 1); y > x – 7


5-5 Solving Linear InequalitiesWhen 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 _____________


the points _______ the boundary line are _________________


the points _______ the boundary line are _________________

STEP 1 – HOW SHOULD I DRAW THE

BOUNDARY LINE?

STEP 2 – HOW SHOULD I SHADE?

𝒚 ≤𝒐𝒓 𝒚 ≥

are solutions

solid

𝒚<𝒐𝒓 𝒚>¿

are not solutions

dashed

𝒚>𝒐𝒓 𝒚 ≥ 𝒚<𝒐𝒓 𝒚 ≤above below

solutions of the inequality solutions of the inequality



Graphing Linear Inequalities

Step 1 Solve the inequality for y (slope-intercept form).

Step 2Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >.

Step 3Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.



Graph the solutions of the linear inequality.

Example 2A: Graphing Linear Inequalities in Two Variables

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.



Example 2A Continued

Substitute (0, 0) for (x, y) because it is not on the boundary line.Check y 2x – 3

0 2(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.


y 2x – 3



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

Helpful Hint




Example 2B: Graphing Linear Inequalities in Two Variables

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.



Step 3 The inequality is >, so shade above the line.

Example 2B Continued

Graph the solutions of the linear inequality.5x + 2y > –8



Example 2B Continued

Substitute ( 0, 0) for (x, y) because it is not on the boundary line.

The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

Check

y > x – 4

0 (0) – 4

0 –40 –4>

Graph the solutions of the linear inequality.5x + 2y > –8


5-5 Solving Linear InequalitiesTry on your own!!!


2x – y – 4 > 0

http://www.mathnstuff.com/papers/planes.htm

http://www.mathnstuff.com/papers/planes.htm



Ada has at most 285 beads to make jewelry. A necklace requires 40 beads, and a bracelet requires 15 beads.

Example 3: Application

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.”



Example 3a Continued

Necklacebeads

braceletbeadsplus

is atmost

285beads.

40x + 15y ≤ 285

Solve the inequality for y.

40x + 15y ≤ 285–40x –40x

15y ≤ –40x + 285Subtract 40x from

both sides.

Divide both sides by 15.



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 ≤.



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.

Example 3b Continued



c. Give two combinations of necklaces and bracelets that Ada could make.

Example 3c

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)



Write an inequality to represent the graph.Example Together

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



Try on your own!!!

Write an inequality to represent the graph.

y-intercept: slope:



HOMEWORK

PG. 364-366

#12-21, 30-40(evens), 41, 42

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