Chapter 1.4, Problem 56E

To determine

To complete: the give statement with always, sometimes or never.

Answer to Problem 56E

For any real number p, the equation $|x-p|=4$ will always have two solutions.

Explanation of Solution

Given:

For any real number p, the equation $|x-p|=4$ will -----------have two solutions.

Calculation:

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

  $ax+b=c\text{ or }ax+b=-c$

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 $|x-p|=4$ will always have two solutions because the right-hand side of the equation is positive.

For any real number p, the equation $|x-p|=4$ will always have two solutions.

Conclusion:

For any real number p, the equation $|x-p|=4$ will always have two solutions.