Transcript Absolute Value Equations and Inequalities

Absolute Value
Equations and
Inequalities
Section 3-7 Part 2
Goals
Goal
• To solve inequalities
involving absolute value.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies the
goals to different and more
complex problems.
Vocabulary
• None
Absolute Value Inequalities
• When an inequality contains an
absolute-value expression, it can be
written as a compound inequality.
• The inequality |x| < 5 describes all real
numbers whose distance from 0 is less
than 5 units.
• The solutions are all numbers between
–5 and 5, so |x|< 5 can be rewritten as
–5 < x < 5, or as x > –5 AND x < 5.
“Less Than” Absolute Value
Inequalities
Example: “Less Than” Absolute
Value Inequalities
Solve the inequality and graph the solutions.
|x|– 3 < –1
Since 3 is subtracted from |x|, add 3 to both
sides to undo the subtraction.
|x|– 3 < –1
+3 +3
|x| < 2
x > –2 AND x < 2
2 units
–2
–1
Write as a compound inequality.
The solution set is {x: –2 < x < 2}.
2 units
0
1
2
Example: “Less Than” Absolute
Value Inequalities
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x – 1 ≥ –2 AND x – 1 ≤ 2
+1 +1
+1 +1
x ≥ –1 AND
Write as a compound inequality.
Solve each inequality.
x≤3
Write as a compound inequality.
The solution set is
{x: –1 ≤ x ≤ 3}.
–3
–2
–1
0
1
2
3
Procedure Solving “Less Than”
Inequalities
Solving Absolute Inequalities:
1. Isolate the absolute value.
2. Break it into 2 inequalities (and
statement) – one positive and the
other negative reversing the sign.
3. Solve both inequalities.
4. Check both answers.
Helpful Hint
Just as you do when solving absolute-value equations,
you first isolate the absolute-value expression when
solving absolute-value inequalities.
Your Turn:
Solve the inequality and graph the solutions.
2|x| ≤ 6
2|x| ≤ 6
2
2
|x| ≤ 3
Since x is multiplied by 2, divide both sides
by 2 to undo the multiplication.
Write as a compound inequality.
The solution set is {x: –3 ≤ x ≤ 3}.
x ≥ –3 AND x ≤ 3
3 units
–3
–2
–1
3 units
0
1
2
3
Your Turn:
Solve each inequality and graph the solutions.
|x + 3|– 4.5 ≤ 7.5
|x + 3|– 4.5 ≤ 7.5
+ 4.5 +4.5
Since 4.5 is subtracted from |x + 3|,
add 4.5 to both sides to undo
the subtraction.
|x + 3| ≤ 12
x + 3 ≥ –12 AND x + 3 ≤ 12
–3 –3
–3 –3
Write as a compound inequality.
The solution set is
{x: –15 ≤ x ≤ 9}.
x ≥ –15 AND x ≤ 9
–20
–15 –10
–5
0
5
10
15
“Greater Than” Absolute Value
Inequalities
• The inequality |x| > 5 describes all
real numbers whose distance from
0 is greater than 5 units.
• The solutions are all numbers less
than –5 or greater than 5.
• The inequality |x| > 5 can be
rewritten as the compound
inequality x < –5 OR x > 5.
“Greater Than” Absolute Value
Inequalities
Example: “Greater Than” Absolute
Value Inequalities
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
|x| + 14 ≥ 19
– 14 –14
|x|
≥ 5
x ≤ –5 OR x ≥ 5
Since 14 is added to |x|, subtract 14 from both
sides to undo the addition.
Write as a compound inequality. The solution
set is {x: x ≤ –5 OR x ≥ 5}.
5 units
–10 –8 –6 –4 –2
5 units
0
2
4
6
8 10
Example: “Greater Than” Absolute
Value Inequalities
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
Since 3 is added to |x + 2|, subtract 3
from both sides to undo the addition.
3 + |x + 2| > 5
–3
–3
|x + 2| > 2
x + 2 < –2 OR x + 2 > 2
–2 –2
–2 –2
Write as a compound inequality. Solve
each inequality.
Write as a compound inequality.
x < –4 OR x > 0
–10 –8 –6 –4 –2
The solution set is
{x: x < –4 or x > 0}.
0
2
4
6
8 10
Procedure Solving “Greater Than”
Inequalities
Solving Absolute Inequalities:
1. Isolate the absolute value.
2. Break it into 2 inequalities (or
statement) – one positive and the
other negative reversing the sign.
3. Solve both inequalities.
4. Check both answers.
Your Turn:
Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
|x| + 10 ≥ 12
– 10 –10
|x|
Since 10 is added to |x|, subtract 10 from
both sides to undo the addition.
≥ 2
x ≤ –2 OR x ≥ 2
Write as a compound inequality. The solution set
is {x: x ≤ –2 or x ≥ 2}.
2 units 2 units
–5 –4 –3 –2 –1
0
1
2
3
4
5
Your Turn:
Solve the inequality and graph the solutions.
Since is added to |x + 2 |, subtract
from both sides to undo the addition.
Write as a compound inequality.
Solve each inequality.
OR
x ≤ –6
x≥1
–7 –6 –5 –4 –3 –2 –1
Write as a compound
inequality. The solution
set is {x: x ≤ –6 or x ≥ 1}
0
1
2
3
Example: Application
A pediatrician recommends that a baby’s bath water be
95°F, but it is acceptable for the temperature to vary from
this amount by as much as 3°F. Write and solve an
absolute-value inequality to find the range of acceptable
temperatures. Graph the solutions.
Let t represent the actual water temperature.
The difference between t and the ideal
temperature is at most 3°F.
t – 95
≤
3
Example: Continued
t – 95
≤
3
|t – 95| ≤ 3
t – 95 ≥ –3 AND t – 95 ≤ 3
+95 +95
+95 +95
t
≥ 92 AND t
90
Solve the two
inequalities.
≤ 98
92
94
96
98
100
The range of acceptable temperature is 92 ≤ t ≤ 98.
Your Turn:
A dry-chemical fire extinguisher should be pressurized to
125 psi, but it is acceptable for the pressure to differ from
this value by at most 75 psi. Write and solve an absolutevalue inequality to find the range of acceptable pressures.
Graph the solution.
Let p represent the desired pressure.
The difference between p and the ideal pressure
is at most 75 psi.
p – 125
≤
75
Your Turn: Continued
p – 125
≤
75
|p – 125| ≤ 75
p – 125 ≥ –75 AND p – 125 ≤ 75
+125 +125
+125 +125
p
≥
50 AND p
25
50
Solve the two
inequalities.
≤ 200
75
100 125 150 175
200
225
The range of pressure is 50 ≤ p ≤ 200.
Solutions to Absolute Value Inequalities
• When solving an absolute-value
inequality, you may get a statement that
is true for all values of the variable.
• In this case, all real numbers are
solutions of the original inequality.
• If you get a false statement when
solving an absolute-value inequality, the
original inequality has no solutions. Its
solution set is ø.
Example:
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4|– 5 > – 8
+5 +5
|x + 4|
> –3
Add 5 to both sides.
Absolute-value expressions are
always nonnegative. Therefore,
the statement is true for all real
numbers.
The solution set is all real numbers.
Example:
Solve the inequality.
|x – 2| + 9 < 7
|x – 2| + 9 < 7
–9 –9
|x – 2|
< –2
Subtract 9 from both sides.
Absolute-value expressions are
always nonnegative. Therefore,
the statement is false for all values
of x.
The inequality has no solutions. The solution set is ø.
Remember!
An absolute value represents a distance, and
distance cannot be less than 0.
Your Turn:
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11
+9 ≥ +9
|x|
≥ –2
Add 9 to both sides.
Absolute-value expressions are
always nonnegative. Therefore,
the statement is true for all real
numbers.
The solution set is all real numbers.
Your Turn:
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8
4
4
|x – 3.5| ≤ –2
Divide both sides by 4.
Absolute-value expressions are
always nonnegative. Therefore,
the statement is false for all values
of x.
The inequality has no solutions. The solution set is ø.
SUMMARY
Absolute Value Inequalities (< , )
Inequalities of the Form < or  Involving Absolute Value
If a is a positive real number and if u is an algebraic
expression, then
|u| < a is equivalent to u < a and u > -a
|u|  a is equivalent to u  a and u > -a
Note: If a = 0, |u| < 0 has no real solution, |u|  0 is equivalent
to u = 0. If a < 0, the inequality has no real solution.
Absolute Value Inequalities (> , )
Inequalities of the Form > or  Involving Absolute Value
If a is a positive real number and u is an algebraic
expression, then
|u| > a is equivalent to u < – a or u > a
|u|  a is equivalent to u  – a or u  a.
Procedure
Solving Absolute Inequalities:
1. Isolate the absolute value.
2. Break it into 2 inequalites (“< , ≤ and
statement” – “> , ≥ or statement”) –
one positive and the other negative
reversing the sign.
3. Solve both inequalities.
4. Check both answers.
Solving an Absolute-Value Equation
Recall that |xx is
| isthe
thedistance
distancebetween
betweenxxand
and0.0.IfIf x
| x 
| 8,8,then
then
any number between 8 and 8 is a solution of the inequality.
8 7 6 5 4 3 2 1
0
1
2
3
4
5
6
You can use the following properties to solve
absolute-value inequalities and equations.
7
8
SOLVING ABSOLUTE-VALUE INEQUALITIES
SOLVING ABSOLUTE-VALUE INEQUALITIES
| ax  b |  c
means
ax  b  c
and
a x  b   c.
a number,
means valueais
| a xWhen
 b | an
c absolute
x less
b than
c
and
a x the
b   c.
inequalities are connected by and. When an absolute
value is greater than a number, the inequalities are
connected by or.
| ax  b |  c
means
ax  b  c
or
a x  b   c.
| ax  b |  c
means
ax  b  c
or
a x  b   c.
Solving an Absolute-Value Inequality
Solve | x  4 | < 3
x  4 IS POSITIVE
x  4 IS NEGATIVE
|x4|3
|x4|3
x  4  3
x  4  3
x7
x1
Reverse
inequality
symbol.
The solution is all real numbers greater than 1 and
less than
7.
This can be written as 1  x  7.
–10
–8 –6 –4
–2
0
2
4
6
8
10
Solving an Absolute-Value Inequality
Solve
 1 | 3  6 and graph
2x + 1| 2x
IS POSITIVE
2x +the
1 ISsolution.
NEGATIVE
| 2x  1 |  3  6
| 2x  1 | 3  6
2x + 1 IS POSITIVE
| 2x|2x1 | 13|  69
2x + 1 IS NEGATIVE
1 |  69
| 2x|2x1 | 3
2x  1  9
| 2x  1 |  9
2x  10
2x  8
2x  1  9
2x  1  +9
x4
x  5
2x  10
2x  8
The solution is all real numbers greater than or equal
x4
x  5
to 4 or less than or equal to  5. This can be written as
the compound inequality
x   5 or x  4.
Reverse
2x11|  +9
| 2x
9
inequality symbol.
6 5 4 3 2 1
0
1
2
3
4
5
6
Joke Time
• What's the difference between chopped beef
and pea soup?
• Everyone can chop beef, but not everyone
can pea soup!
• What do you get when you cross an elephant
and a rhino?
• el-if-i-no
• What’s a quick way to double your money?
• You fold it!
Assignment
3.7 pt 2 Exercises Pg. 228 – 229: #6 – 42
even