Q56. - Algebra 2

Question

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

Notice that each equation you wrote in exercise 55 has two cases. For each equation, write two other equations taking into consideration the case where $x+1\ge 0$ and the case where $x+1<0$ .

Step-by-Step Solution

Answer

When $x+1\ge 0$ equations from exercise 55 becomes $x+1+2=x+4$ and $x+1+2=-\left(x+4\right)$

When $x+1<0$ equation from exercise 55 becomes $-\left(x+1\right)+2=x+4$and $-\left(x+1\right)+2=-\left(x+4\right)$

Step 1 - Define absolute value

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

Step 2 - Case when x + 1 &#8805; 0

Using definition of absolute value, $\left|x+1\right|=x+1$ when $x+1\ge 0$ 

Thus, equations from exercise 55 becomes $x+1+2=x+4$ and $x+1+2=-\left(x+4\right)$

Step 3 - Case when x + 1 &#60; 0

Using definition of absolute value, $\left|x+1\right|=-\left(x+1\right)$ when $x+1<0$

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