Chapter 1.4, Problem 57E

To determine

Tofind:the reason absolute value equations can have no solution, one solution or two solutions.

Answer to Problem 57E

If the absolute value is greater than zero come up there will be one or two solutions.

Explanation of Solution

Given:

Absolute value equations.

Concept used:

Absolute value or modulus of a real number $x$ ,denoted that $\left|x\right|$ , is the non-negative value of $x$ without regard to its sign.

Mathematically,

  $\left|x\right|=x.$

If domain $x$ is positive.

  $\left|x\right|=-x.$

If domain $x$ is negative.

Calculation:

The absolute value of an expression must be greater than or equal to $0$ .

The equation has no solution if it is equal to negative number.

Example:

  $\begin{array}{l}\left|x+3\right|=-3.\\ \text{or}\\ \left|x+3\right|+6=3.\\ \left|x+3\right|=-3.\end{array}$

If the absolute value is compared to $0$ , there will be one solution.

  $\begin{array}{l}\left|x+3\right|+6=3.\\ \Rightarrow \left|x+3\right|=-3.\end{array}$

If the absolute value is greater than zero come up there will be one or two solutions.

  $\begin{array}{l}\left|x+3\right|=6.\\ x+3=6.\\ x=3.\\ and\\ x+3=-6.\\ x=-9.\end{array}$

The equation has two solution.

Another example:

  $\begin{array}{l}\left|x-3\right|=\left|x+2\right|.\\ x-3=x+2.\\ -3\ne 2.\\ \text{and}\\ x-3=-\left(x+2\right).\\ x=\frac{1}{2}.\end{array}$

Hence, If the absolute value is greater than zero come up there will be one or two solutions.