4.1Solving Linear Inequalities

11/2/2012

You have learned how to solve equations with 1 variable.Ex. x + 3 = 7 -3 -3 x = 4

Ex. x - 5 = 2 +5 +5 x = 7

Ex. 3x = 12 3 3 x = 4

Ex. y = 6 2 y = 12

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To solve inequality in 1 variable

All the rules apply for solving equations in 1 variable except when dividing or multiplying both sides by a negative number. Rule: When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

Example 1 Inequality with a Variable on One Side

Solve the inequality.

a. b.>x 4– 6– – ≥5y 2 13–+

SOLUTION

a. >x 4– 6– Write original inequality.

>x 2– Add 4 to each side.

ANSWER

The solution is all real numbers greater than 2.–

+ 4 + 4

Example 1 Inequality with a Variable on One Side

b. – ≥5y 2 13–+ Write original inequality.

– ≥5y 15– Subtract 2 from each side.

ANSWER

The solution is all real numbers less than or equal to 3.

Divide each side by 5 and reverse the inequality.

– y ≤ 3

-2 -2

-5 -5

Graphing the solution of an inequality in 1 variable.

For > or <, doughnut ( )For ≤ or ≥, solid circle ( )

Then shade the side of the number line according to which way the inequality symbol is pointing when the variable is on the left side

Ex. x > -1

Ex. x ≤ 3

Example 2 Inequality with a Variable on Both Sides

Solve . Graph the solution.

Add 2x to each side.<7 2x– 1

<7 4x 1 2x––

SOLUTION

Write original inequality.<7 4x 1 2x––

<2x– 6– Subtract 7 from each side.

>x 3 Divide each side by 2 and reverse the inequality.

–

+2x +2x

-2 -2

Example 2 Inequality with a Variable on Both Sides

ANSWER

The solution is all real numbers greater than 3. The graph is shown at the bottom.

2 3 4 511– 0

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Inequality

<x 3 8+1.

5≤4 x–2.

ANSWER <x 5

210 3 4 5 6

ANSWER ≥x 1–

0 22–

Checkpoint

Solve the inequality. Then graph your solution.

– x>2x 34.

Solve an Inequality

2>2x 1–3.

3x >ANSWER

210 3 4 5 6

x >2

3ANSWER

1 20

Example 3 Use a Simple Inequality

Amusement Park Admission to an amusement park costs $9 and each ride ticket costs $1.50. The total amount A in dollars spent is given by where t is the number of ride tickets. Use an inequality to describe the number of ride tickets you can buy if you have at most $45 to spend at the park.

=A 1.5t + 9

SOLUTION The most money you can spend is $45.A 45≤

Substitute for A.45≤1.5t + 9 1.5t + 9

Subtract 9 from each side.36≤1.5t

Divide each side by 1.5.24≤t

ANSWER You can buy up to 24 ride tickets.

COMPOUND INEQUALITY2 simple inequalities joined by the word “and” or the word “or”. Ex: ANDAll real numbers greater than or equal to -2 AND less than 1.

Ex: ORAll real numbers less than -1 OR greater than or equal to 2

-2≤ x < 1

x < -1 or x ≥ 2

Example 4 Solve an “Or” Compound Inequality

Solve or3x + 2 8< 2x 9 3.– >

SOLUTION

Solve each part separately.

FIRST INEQUALITY SECOND INEQUALITY

3x + 2 8< 2x 9 3– >Write first inequality.

Write second inequality.

3x 6< 2x 12>Subtract 2 from each side.

Add 9 to eachside.

x 2< x 6>Divide each sideby 3.

Divide each side by 2.

Checkpoint

Solve the inequality. Then graph your solution.

Solve Compound Inequalities

5. 4 x< + 5 7<

6. 3x≤ + 81– 8≤

ANSWER x 2<1 <–

2– 20

ANSWER x≤3– 0≤

4– 2– 0..

Checkpoint

Solve the inequality. Then graph your solution.

Solve Compound Inequalities

7. x + 3 4 or ≤ x 6– ≥ 1–

ANSWER x 1 or≤ x 5≥

420. .

8. 2x 6 0<– –x 4 or– >

ANSWER x 4 or–< x 3–>

06– 4– 2–

Homework4.1 p.175 #14-22 even, 23-28all, 30-38even, 40-44all, 50-56even