Chapter 1.4, Problem 61E

To determine

Tofind:the number of solutions does the equation have and give reason for that.

Answer to Problem 61E

It means that the equation has two solutions.

Explanation of Solution

Given:

  $\begin{array}{l}a\left|x+b\right|+c=d.\\ a>0,c=d.\end{array}$

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:

According to the given:

  $\begin{array}{l}a\left|x+b\right|+c=d.\\ a>0,c=d.\end{array}$

The equation can be written as:

  $\begin{array}{l}a\left|x+b\right|=d-c.\\ a\left|x+b\right|=0.\end{array}$

The equation has one solution:

  $\begin{array}{l}x+b=0.\\ x=-b.\end{array}$

According to the given equation:

  $\begin{array}{l}a\left|x+b\right|+c=d.\\ a>0,c>d.\end{array}$

The equation can be written as:

  $\begin{array}{l}a\left|x+b\right|=d-c.\\ \left|x+b\right|=\frac{d-c}{a}.\end{array}$

Since, $c>d$ it means $d-c<0$ .

Since, $a<0$ also it means that the right side is positive.

Hence, it means that the equation has two solutions.