Absolute Value Inequalities_2

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Solving an absolute value inequality problem is similar to solving an absolute value equation.

Start by isolating the absolute value on one side of the inequalitysymbol, then follow the rules below:

If the symbol is > (or >): (or)

If a > 0, then the solutions to

are x> a or x< -a.

If a< 0, all real numbers

will satisfy .

Think about it: absolute value is alwayspositive (or zero), so, of course, it is greater than

anynegative number.

If the symbol is < (or < ): (and)

If a> 0, then the solutions to

are x< a and x> -a.Also written: -a -aCase 2: Write the problem withoutthe absolute value sign, reverse theinequality, negate the value NOT

under the absolute value, and solvethe inequality.

Your graphing calculator can be used to solve absolute valueinequalities and/or double check your answers.

How to use yourTI-83+ graphingcalculator withabsolute valueinequalities.

Click calculator.

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Example 1: (solving with "greater than")

Solve:Case 1:

or

Case 2:

x< 15 or x> 25

Note that there are two parts to the

solution and that the connectingword is "or".

Example 2: (solving with "less than or equal to")

Solve:

Case 1:

and

Case 2:Note that there are two parts to the

solution and that the connecting

word is "and".

also written as:

Example 3: (isolating the absolute value first)

Solve:

Case 1:

and

Case 2: Note that the absolute value is

isolated before the solution begins.

also written as:


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Example 4: (compound inequalities)Separate a compound inequality into two

separate problems.

Solve:

Case 1:

orCase 2: Case 1:

andCase 2:

x> 4 or x< -6 -8


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Case 1:

and

Case 2: (or zero). It isNEVER < -6.

No values work!

x < -7 and x> 5 ???Answer:

(the empty set)

Example 7: (word problem)

At the Brooks Graphic Company, the average starting

salary for a new graphic designer is $37,600, but the actualsalary could differ from the average by as much $2590.

The absolute value

represents the set of all pointsxthat are less than bunits away

from a.

b.)Case 1:

andCase 2:

$35,010 < x < $40,190

a.) Write an absolute value inequalityto describe this situation.

b.) Solve the inequality to find therange of the starting salaries.

Solution: a.)

|the difference between the average and the salary| < $2590


09/09/2013http://www regentsprep org/regents/math/algtrig/ATE2/absinequal htm

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