1-3: Absolute Value Equations and Inequalities
Objectives:
1. Write, solve, and
graph absolute value
equations
2. Solve and graph
absolute value
inequalities
Assignment:
• P. 14: 6-14
• P. 16: 22-28
• Challenge Problems
Absolute Value
The absolute
value of a
number is its
distance from
zero on a real
number line.
Absolute value is
always positive.
Absolute Value Equations
An absolute value equation is an equation
containing the absolute value of an algebraic
expression.
Absolute Value Equations
An absolute value equation is an equation
containing the absolute value of an algebraic
expression.
𝑎𝑥 + 𝑏 = 𝑐
𝑎𝑥 + 𝑏 = 𝑐
𝑎𝑥 + 𝑏 = −𝑐
When solving
an absolute
value
equation, first
isolate the
absolute
value bars.
Example A
Solve 2 𝑥 − 1 − 5 = 1. Graph the solutions on a
number line.
Exercise 1
Solve 4𝑥 + 10 = 6𝑥. Graph the solutions
on a number line.
Extraneous Solutions
In the course of solving an
absolute value equation,
one of the solutions may
not actually satisfy the
original equation.
This an extraneous
solution. Get rid of it;
it’s no good!
Example B
The temperature of the wave pool at Sapphire Island
can vary up to 4.5° F from the target temperature of
82° F. Write and solve an absolute value equation to
find the temperature extremes of the wave pool.
Absolute Value Inequalities
Follow me here:
|x| = 5 means the distance from zero equals
5.
-5
0
5
|x| ≤ 5 means the distance from zero is less
than or equal to 5.
-5
0
−5 ≤ x ≤ 5
5
Absolute Value Inequalities
Follow me here:
|x| = 5 means the distance from zero equals
5.
-5
0
5
|x| ≥ 5 means the distance from zero is
greater than or equal to 5.
-5
0
x ≤ −5 or x ≥ 5
5
Absolute Value Inequalities
If the general methods from the previous
slides are incomprehensible, you could just
memorize these.
Absolute Value Inequalities
If the general methods from the previous
slides are incomprehensible, you could just
memorize these.
Less thAND
Absolute Value Inequalities
If the general methods from the previous
slides are incomprehensible, you could just
memorize these.
GreatOR
Example C
Solve each inequality. Graph the solutions on a
number line.
1.
2𝑥 + 3 + 1 > 6
Example C
Solve each inequality. Graph the solutions on a
number line.
2.
3𝑥 − 1 + 5 < 7
Tolerance A
A professional baseball
should weigh 5.125 ounces,
with a tolerance of 0.125
ounces. (Tolerance is the
maximum deviation from
the ideal measurement.)
Write and solve an absolute
value inequality that
describes the acceptable
weights of a baseball.
Tolerance B
You have found that your
new winter coat is
comfortable to wear
when the outdoor
temperature is between
10°F and 42°F,
inclusive. Write an
absolute value inequality
for this temperature
range in degrees
Fahrenheit.
Tolerance C
The temperature of the wave pool at
Sapphire Island can vary up to 4.5° F from
the target temperature of 82° F. Write and
solve an absolute value inequality to find the
range of temperature of the wave pool.
1-3: Absolute Value Equations and Inequalities
Objectives:
1. Write, solve, and
graph absolute value
equations
2. Solve and graph
absolute value
inequalities
Assignment:
• P. 14: 6-14
• P. 16: 22-28
• Challenge Problems