Transcript Chapter 1.7 Solve Absolute Value Equations and Inequalities

Chapter 1.7 Solve Absolute Value
Equations and Inequalities
Analyze Situations using algebraic
symbols; Use models to understand
relationships
Chapter 1.7 Solve Absolute Value
Equations and Inequalities
Chapter 1.7 Solve Absolute Value
Equations and Inequalities
Chapter 1.7 Solve Absolute Value
Equations and Inequalities
EXAMPLE 2
Solve an absolute value equation
Solve |5x – 10 | = 45.
SOLUTION
| 5x – 10 | = 45
Write original equation.
5x – 10 = 45 or 5x – 10 = –45 Expression can equal 45 or –45 .
5x = 55 or
5x = –35
x = 11 or
x = –7
Add 10 to each side.
Divide each side by 5.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
4. |3x – 2| = 13
ANSWER
The solutions are 5 and
-
11
3
.
Chapter 1.7 Solve Absolute Value
Equations and Inequalities
EXAMPLE 4
Solve an inequality of the form |ax + b| > c
Solve |4x + 5| > 13. Then graph the solution.
SOLUTION
The absolute value inequality is equivalent to
4x +5 < –13 or 4x + 5 > 13.
First Inequality
4x + 5 < –13
4x < –18
x<– 9
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Second Inequality
Write inequalities.
4x + 5 > 13
Subtract 5 from each side.
4x > 8
Divide each side by 4.
x>2
EXAMPLE 4
Solve an inequality of the form |ax + b| > c
ANSWER The solutions are all real numbers less than
– 9 or greater than 2. The graph is shown below.
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GUIDED PRACTICE
for Example 4
Solve the inequality. Then graph the solution.
7. |x + 4| ≥ 6
ANSWER
x < –10 or x > 2 The graph is shown below.
GUIDED PRACTICE
for Example 4
Solve the inequality. Then graph the solution.
8. |2x –7|>1
ANSWER
x < 3 or x > 4 The graph is shown below.
GUIDED PRACTICE
for Example 4
Solve the inequality. Then graph the solution.
9. |3x + 5| ≥ 10
ANSWER
x < –5 or x > 1 23 The graph is shown below.
EXAMPLE 5
Solve an inequality of the form |ax + b| ≤ c
Baseball A professional baseball should weigh 5.125
ounces, with a tolerance of 0.125 ounce. Write and
solve an absolute value inequality that describes the
acceptable weights for a baseball.
SOLUTION
STEP 1
Write a verbal model. Then write an
inequality.
EXAMPLE 5
STEP 2
Solve an inequality of the form |ax + b| ≤ c
Solve the inequality.
|w – 5.125| ≤ 0.125
– 0.125 ≤ w – 5.125 ≤ 0.125
5 ≤ w ≤ 5.25
Write inequality.
Write equivalent compound
inequality.
Add 5.125 to each expression.
ANSWER
So, a baseball should weigh between 5 ounces and
5.25 ounces, inclusive. The graph is shown below.
GUIDED PRACTICE
for Examples 5 and 6
Solve the inequality. Then graph the solution.
10. |x + 2| < 6
ANSWER
–8 < x < 4
The solutions are all real numbers less than – 8 or
greater than 4. The graph is shown below.
GUIDED PRACTICE
for Examples 5 and 6
Solve the inequality. Then graph the solution.
11. |2x + 1| ≤ 9
ANSWER
–5 ≤ x ≤ 4
The solutions are all real numbers less than –5 or
greater than 4. The graph is shown below.
GUIDED PRACTICE
for Examples 5 and 6
Solve the inequality. Then graph the solution.
12. |7 – x| ≤ 4
ANSWER
3 ≤ x ≤ 11
The solutions are all real numbers less than 3 or
greater than 11. The graph is shown below.
Solve |x – 5| = 7. Graph the solution.
SOLUTION
|x– 5|=7
x– 5=–7
Write original equation.
or x – 5 = 7
Write equivalent equations.
x = 5 – 7 or
x=5+7
Solve for x.
x = –2
x = 12
Simplify.
or
The solutions are –2 and 12. These are the values
of x that are 7 units away from 5 on a number line.
The graph is shown below.
EXAMPLE 3
Check for extraneous solutions
Solve |2x + 12 | = 4x. Check for extraneous solutions.
SOLUTION
| 2x + 12 | = 4x
Write original equation.
2x + 12 = 4x or 2x + 12 = – 4x Expression can equal 4x or – 4 x
12 = 2x or 12 = –6x
6=x
or –2 = x
Add –2x to each side.
Solve for x.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
2. |x – 3| = 10
ANSWER
The solutions are –7 and 13. These are the values
of x that are 10 units away from 3 on a number line.
The graph is shown below.
10
–7 –6–5–4 –3 –2–1 0 1 2
10
3 4
5
6 7 8 9 10 11 12 13
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
3. |x + 2| = 7
ANSWER
The solutions are –9 and 5. These are the values of
x that are 7 units away from – 2 on a number line.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
5.
|2x + 5| = 3x
ANSWER
The solution of is 5. Reject 1 because it is an
extraneous solution.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
6.
|4x – 1| = 2x + 9
ANSWER
The solutions are –1 1 and 5.
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Chapter 1.7 Solve Absolute Value
Equations and Inequalities
Homework Pg. 55
22-38 evens, 44-52 evens, 66-69