Question

Determine whether the following statement is sometimes, always, or never true. Explain.

For all real numbers $a$and$b,a\ne 0$, the equation$\left|ax+b\right|=0$ will have one solution.

Short answer

Given statement is always true.

Since, $\left|ax+b\right|=0$is true when ax+b=0 which is true only for $x=-\frac{b}{a}$

So given equation is always true.

Step-by-step solution

- Define absolute value

Absolute value of a number is its distance from 0 on the number line.

- Meaning of ax+b=0

Distance of expression ax+b is 0 from origin.

So ax+b=0

- Find ax+b=0 has one or more solution

As ax+b=0 is a linear equation which has only one solution namely $x=-\frac{b}{a}$

So given statement is always true.