Question

For Exercises 55-58, consider the equation $\left|x+1\right|+2=\left|x+4\right|$.

55. To solve this equation we must consider the case where $x+4\ge 0$and the case where $x+4<0$. Write the equations for each of these cases.

Short answer

When $x+4\ge 0$ given equation becomes $\left|x+1\right|+2=x+4$

When $x+4<0$ given equation becomes $\left|x+1\right|+2=-\left(x+4\right)$

Step-by-step solution

- Define absolute value

For a real number a, $\left|a\right|=a$ if $a\ge 0$ and $\left|a\right|=-a$ if $a<0$

- Case when x+4≥0

Using definition of absolute value, $\left|x+4\right|=x+4$ when $x+4\ge 0$

Thus, given equation becomes $\left|x+1\right|+2=x+4$

- Case when x+4<0

Using definition of absolute value, $\left|x+4\right|=-\left(x+4\right)$when x+4<0

Thus, given equation becomes $\left|x+1\right|+2=-\left(x+4\right)$