Chapter 1.4, Problem 56E

To determine

Tofind:the statement always, sometimes or never that can be filled in the blank.

Answer to Problem 56E

$p$ is any real number, then the equation $\left|x-p\right|=4$ will have always two solutions.

Explanation of Solution

Given:

  $\left|x-4\right|=p$ .

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:

In solving absolute value equations, remember that $\left|ax+b\right|=c$ provided that $c\ge 0$ will either be:

  $ax+b=c\text{ or }ax+b=-c$ .

Subtracting both the side by $b$ .

Note that when $c<0$ , the equation has no solution because absolute values are always positive.

Since, $p$ is any real number, then the equation $\left|x-p\right|=4$ will always have two solutions because, the right band side of the equation is positive.

Hence, $p$ is any real number, then the equation $\left|x-p\right|=4$ will always have two solutions because, the right band side of the equation is positive.