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1.7 Solve Absolute Value Equations and Inequalities
Goal  Solve absolute value equations and inequalities.
Your Notes
VOCABULARY
Absolute value
The absolute value of a number x, written | x|, is the distance the number is from 0 on a
number line.
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Extraneous solution
An apparent solution that must be rejected because it does not satisfy the original
equation.
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INTERPRETING ABSOLUTE VALUE EQUATIONS
Equation
Meaning
|x| = |x  0| = k
The distance between x and 0 is __k__ .
Graph
Solutions
Equation
Meaning
Graph
Solutions
x  0 = k or x  0 = k
x = _k_ or x = _k_
|x  b| = k
The distance between x and b is _k_.
x  b = k or x b = k
x = _b  k_ or x = _b + k_
Your Notes
Example 1
Solve a simple absolute value equation
Solve |x  3 | = 6. Graph the solution.
|x  3| = 6
Write original equation.
x  3 = _6_ or x  3 = _6_
Write equivalent equations.
x = _3  6_ or x = _3 + 6_
Solve for x.
x = _3_ or
x = _9_
Simplify.
The solutions are _3_ and _9_ .These are the values of x that are _6_ units away from
_3_ on a number line.
Checkpoint Solve the equation. Then graph the solution.
1. |x|= 5
x = 5 or x = 5
2. |x  5| = 2
x = 3 or x = 7
SOLVING AN ABSOLUTE VALUE EQUATION
Use these steps to solve an absolute value equation | a + b | = c where c > 0.
Step 1 Write two equations: ax + b = _c_ or ax + b = _ c__ .
Step 2 _Solve_ each equation.
Step 3 Check each solution in the original _absolute value_ equation.
Your Notes
Example 2
Solve an absolute value equation
Solve |4x + 10 | = 6x. Check for extraneous solutions.
|4x + 10 | = 6x
Write original equation.
4x + 10 = _6x_ or 4x + 10 = _6x_
Expression can equal _6x_ or _6x_ .
10 = _2x_ or
10 = _10x_
Subtract _4x_ from each side.
_5_ = x
or
_1_ = x
Solve for x.
Check the apparent solutions to see if either is extraneous.
CHECK
|4x + 10 | = 6x
|4(_5_) + 10| ≟ 6(_5_)
|_30_| ≟ _30_
_30_ = _30_
|4x + 10 | = 6x
|4( 1_ ) + 10 | ≟ 6(_1_)
|_6_| ≟ _6_
_6_  _6_
Always check your
solutions in the
original equation to
make sure that they
are not extraneous.
The solution is _5_. Reject _1_ because it is an _extraneous_ solution.
Checkpoint Solve the equation. Check for extraneous solutions.
3. |2x + 5| = 11
x = 3 and x = 8
4. |3x + 18| = 6x
x = 6; x = 2 is an extraneous solution.
ABSOLUTE VALUE INEQUALITIES
In the inequalities below, c > 0.
Inequality
Equivalent Form
|ax + b| _<_ c
c < ax + b < c
In the inequalities
shown at the right,
 can replace < and
 can replace > and
the graphs would
have solid dots.
Graph of Solution
|ax + b| _>_ c
ax + b < c or ax + b > c
Your Notes
Example 3
Solve an inequality of the form |ax + b| > c
Solve |2x + 5 | > 3. Then graph the solution.
The absolute value inequality is equivalent to 2x + 5 < _3_ or 2x + 5 > _3_ .
First Inequality
Second Inequality
2x + 5 < _3_
Write inequalities.
2x + 5 > _3_
2x < _8_
Subtract _5_ from each side.
2x > _2_
2x < _4_
Divide each side by _2_ .
x > _1_
The solutions are all real numbers less than _4_ or greater than _1_.
Example 4
Solve an inequality of the form | ax + b |  c
Solve |x  1.5|  4.5. Then graph the solution.
x  1.5  4.5
Write inequality.
_4.5_  x  1.5  _4.5_ Write equivalent compound inequality.
_3.0_  x  _6.0_
Add _1.5_ to each expression.
The solution is between _3_ and _6_ , inclusive.
Checkpoint Solve the inequality. Then graph the solution.
5. |x  2|  7
x  5 or x  9
6. |4x  1| < 9
2 < x < 2.5
Homework
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