Section A.4
Absolute Value Equations
and Inequalities
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Solving Absolute Value Equations
Solve the following equations involving absolute value.
x 5
Solution
x=5
or
3x  4  5
Solution 3x  4  5
3x  9
x3
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x = −5
or
3x  4  5
3x  1
1
x
3
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Solving Absolute Value Equations
4 x  1  8
Solution
There is no number that has a negative absolute value.
Therefore, this equation has no solution. (The solution
is  and the equation is a contradiction.)
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Absolute Value Inequalities
Solving Absolute Value Inequalities with < (or ≤ )
For c > 0:
a. If x  c , then −c < x < c.
b. If ax  b  c , then −c < ax + b < c.
The inequalities in a. and b. are also true if < is replaced
by ≤.
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Solving Absolute Value Inequalities
Solve the following absolute value inequalities and
graph the solution sets.
x 6
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Solving Absolute Value Inequalities
x 3 2
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Solving Absolute Value Inequalities
3 2x  7  15
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Solving Absolute Value Inequalities
1
x 9 
2
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Solving Absolute Value Inequalities
2x  4  4  7
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Absolute Value Inequalities
Notes
The expression x > 3 or x < −3 cannot be combined into
one inequality expression. The word or must separate
the inequalities since any number that satisfies one or
the other is a solution to the absolute value inequality.
There are no numbers that satisfy both inequalities.
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Absolute Value Inequalities
Solving Absolute Value Inequalities with > (or ≥ )
For c > 0:
a. If x  c , then x < −c or x > c.
b. If ax  b  c , then ax + b < −c or ax + b > c.
The inequalities in a. and b. are also true if > is replaced
by ≥.
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Solving Absolute Value Inequalities
Solve the following absolute value inequalities and
graph the solution set.
x 5
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Solving Absolute Value Inequalities
4x  3  2
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Solving Absolute Value Inequalities
3x  8  6
There is nothing to do here except observe that no
matter what is substituted for x, the absolute value will
be greater than −6. Absolute value is always
nonnegative (greater than or equal to 0). The solution
to the inequality is all real numbers, so shade the
entire number line. In interval notation, x is in (−∞, ∞).
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Solving Absolute Value Inequalities
2x  5  5  4
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Section A.6
Graphing Systems of Linear
Inequalities
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Solving Systems of Linear Inequalities
Graphically
1.
For each inequality, graph the boundary line and shade the appropriate
half-plane.
2.
Determine the region of the graph that is common to both half-planes
(the region where the shading overlaps).
(This region is called the intersection of the two half-planes and is the
solution set of the system.)
3.
To check, pick one test-point in the intersection and
verify that it satisfies both inequalities.
Note: If there is no intersection, then the system has no solution.
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Graphing Systems of Linear Inequalities
Graph the points that satisfy the system of inequalities:
x  2

y   x  1
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Solve the system of linear inequalities graphically:
2 x  y  6

 xy4
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Solve the system of linear inequalities graphically:
y  x

y  x  2
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