Power Series and Applications: Background You’ll Need 1

  • Solve inequalities that include absolute values

Absolute Value Inequalities

An absolute value inequality is an equation of the form

[latex]|A| < B,|A|\le B,|A| > B,\text{or }|A|\ge B[/latex],

where [latex]A[/latex], and sometimes [latex]B[/latex], represents an algebraic expression dependent on a variable [latex]x[/latex]. Solving the inequality means finding the set of all [latex]x[/latex] –values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.

absolute value inequality

For an algebraic expression [latex]X[/latex] and [latex]k>0[/latex], an absolute value inequality is an inequality of the form:

[latex]\begin{array}{c} |X| < k \text{ is equivalent to } -k < X < k \\ \text{or} \\ |X| > k \text{ is equivalent to } X < -k \text{ or } X > k \\ \end{array}[/latex]

 

These statements also apply to [latex]|X|\le k[/latex] and [latex]|X|\ge k[/latex].

There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.

Suppose we want to know all possible returns on an investment if we could earn some amount of money within [latex]$200[/latex] of [latex]$600[/latex].

We can solve algebraically for the set of [latex]x-[/latex]values such that the distance between [latex]x[/latex] and [latex]600[/latex] is less than [latex]200[/latex]. We represent the distance between [latex]x[/latex] and [latex]600[/latex] as [latex]|x - 600|[/latex], and therefore,

[latex]|x - 600|\le 200[/latex]
or
[latex]\begin{array}{c}-200\le x - 600\le 200\\ -200+600\le x - 600+600\le 200+600\\ 400\le x\le 800\end{array}[/latex]

This means our returns would be between [latex]$400[/latex] and [latex]$800[/latex].

To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

Describe all values [latex]x[/latex] within a distance of [latex]4[/latex] from the number [latex]5[/latex].

A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.
Absolute value shows distance, so both directions from 5 are included

We want the distance between [latex]x[/latex] and [latex]5[/latex] to be less than or equal to [latex]4[/latex]. We can draw a number line to represent the condition to be satisfied.The distance from [latex]x[/latex] to [latex]5[/latex] can be represented using an absolute value symbol, [latex]|x - 5|[/latex]. Write the values of [latex]x[/latex] that satisfy the condition as an absolute value inequality.

[latex]|x - 5|\le 4[/latex]

We need to write two inequalities as there are always two solutions to an absolute value equation.

[latex]\begin{array}{lll}x - 5\le 4\hfill & \text{and}\hfill & x - 5\ge -4\hfill \\ x\le 9\hfill & \hfill & x\ge 1\hfill \end{array}[/latex]

If the solution set is [latex]x\le 9[/latex] and [latex]x\ge 1[/latex], then the solution set is an interval including all real numbers between and including [latex]1[/latex] and [latex]9[/latex].

So, [latex]|x - 5|\le 4[/latex] is equivalent to [latex]\left[1,9\right][/latex] in interval notation.

Solve the following:

[latex]|x - 1|\le 3[/latex]

Given the equation:

[latex]y=-\frac{1}{2}|4x - 5|+3[/latex],

determine the [latex]x[/latex]-values for which the [latex]y[/latex]-values are negative.