Transcript la1_ch05_06 Solve Absolute Value

5.6 Solve Absolute Value
Inequalities
• You will solve absolute
value inequalities.
• Essential question: How
do you solve absolute
value inequalities?
You will learn how to answer this
question by rewriting absolute
value inequalities as compound
inequalities and solving them.
EXAMPLE
Warm-Up1Exercises
Solve absolute value inequalities
Solve the inequality. Graph your solution.
a. |x|  6
SOLUTION
a. The distance between x and 0 is greater than or
equal to 6. So, x  –6 or x  6.
ANSWER
The solutions are all real
numbers less than or
equal to –6 or greater
than or equal to 6.
EXAMPLE
Warm-Up1Exercises
Solve absolute value inequalities
Solve the inequality. Graph your solution.
b. x  0.5
SOLUTION
The distance between x and 0 is less than or equal
to 0.5. So, –0.5 ≤ x ≤ 0.5.
ANSWER
The solutions are all real
numbers greater than or
equal to –0.5 and less
than or equal to 0.5.
EXAMPLE
4Exercises
Warm-Up
forusing
Example
1
Find a base
the percent
equation
GUIDED
PRACTICE
Solve the inequality. Graph your solution.
1.
|x|  8
ANSWER
–8  x  8
EXAMPLE
4Exercises
Warm-Up
forusing
Example
1
Find a base
the percent
equation
GUIDED
PRACTICE
Solve the inequality. Graph your solution.
2.
|u| < 3.5
ANSWER
–3.5 < u < 3.5
EXAMPLE
4Exercises
Warm-Up
forusing
Example
1
Find a base
the percent
equation
GUIDED
PRACTICE
Solve the inequality. Graph your solution.
3. v > 2
3
ANSWER
2
v < – 32 or v > 3
Warm-Up2Exercises
EXAMPLE
Solve an absolute value inequality
Solve x – 5  7. Graph your solution.
|x–5|7
x – 5  7
x  –2
Write original inequality.
or x – 5  7
or
x  12
Rewrite as compound inequality.
Add 5 to each side.
ANSWER
The solutions are all real numbers less than or equal
to –2 or greater than or equal to 12. Check several
solutions in the original inequality.
Warm-Up3Exercises
EXAMPLE
Solve an absolute value inequality
Solve |–4x – 5| + 3 < 9. Graph your solution.
|–4x – 5| + 3 < 9
|–4x – 5| < 6
–6 < –4x – 5 < 6
Write original inequality.
Subtract 3 from each side.
Rewrite as compound inequality.
–1 < –4x < 11
Add 5 to each expression.
0.25 > x > –2.75
Divide each expression by –4. Reverse
inequality symbol.
–2.75 < x < 0.25
Rewrite in the form a < x < b.
Warm-Up3Exercises
EXAMPLE
Solve an absolute value inequality
ANSWER
The solutions are all real numbers greater than –2.75
and less than 0.25.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Solve the inequality. Graph your solution.
4.
x+3 >8
ANSWER
x < –11
or x > 5
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Solve the inequality. Graph your solution.
5.
2w – 1 < 11
ANSWER
–5 < w < 6
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Solve the inequality. Graph your solution.
6.
3 5m – 6 – 8 <
– 13
ANSWER
–0.2  m  2.6
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
Computers
You are buying a new computer and find 10 models in
a store advertisement. The prices are $890, $750, $650,
$370, $660, $670, $450, $650, $725, and $825.
Find the mean of the computer prices.
You are willing to pay the mean price with an
absolute deviation of at most $100. How many
of the computer prices meet your condition?
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
SOLUTION
STEP 1
Find the mean by dividing the sum of the prices by 10.
Mean =
890 + 750 + 650 + 370 + 660 + 670 + 450 + 650 + 725 + 825
10
6640
= 10
= 664
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
STEP 2
Write and solve an inequality. An absolute deviation
of at most $100 from the mean, $664, is given by the
inequality x – 664 ≤ 100.
x – 664 ≤ 100
– 100 ≤ x – 664 ≤ 100
564 ≤ x ≤ 764
Write absolute value inequality.
Write as compound inequality.
Add 664 to each expression.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
ANSWER
The prices you will consider must be at least $564 and
at most $764. Six prices meet your condition: $750,
$650, $660, $670, $650, and $725.
Warm-Up
Exercises
GUIDED
PRACTICE
7.
for Example 4
WHAT IF? In Example 4, suppose that you are
willing to pay the mean price with an absolute
deviation of at most $75. How many of the computer
prices meet this condition?
ANSWER
5 computer prices
• You will solve absolute
value inequalities.
• Essential question: How
do you solve absolute
value inequalities?
• You can rewrite an absolute
value inequality as a compound
Rewrite the absolute value
inequality. Inequalities using > or
inequality so one side is
≥ can be rewritten using or.
|ax + b |. Then write the
Inequalities using < or ≤ can be
compound inequality as two
rewritten using and.
inequalities joined by or for >
and ≥ , or by and for < and ≤ ,
and solve the inequalities.
Daily
Homework
Quiz
Warm-Up
Exercises
Solve the inequality. Graph your solution.
1.
|x–5|>3
ANSWER
all real numbers less than 2 or greater
than 8
Daily
Homework
Quiz
Warm-Up
Exercises
Solve the inequality.Graph your solution.
2. | x + 3 | + 6  8
ANSWER
all real numbers greater than or equal
to –5 and less than or equal to –1
Daily
Homework
Quiz
Warm-Up
Exercises
3.
The clock in your car has an absolute deviation of
at most 4 minutes after 6 months. After 6 months,
the clock reads 7:38. Write and solve an inequality
to find the possible times.
ANSWER
| x – 7:38 |  4; from 7:34 through 7:42