Question

Solve the following inequality algebraically. 5|x + 5| +3 < 28

Answer 1

Starting with the expression:

`5\lvert x+5\rvert+3<28`

substract 3 from both sides of the inequality:

`5\lvert x+5\rvert<25`

divide both sides by 5:

`\lvert x+5\rvert<5`

There are two cases.

Case 1: x+5 is greater or equal to 0.

In this case, |x+5| = x+5, then:

`\begin{gathered} x+5<5 \\ \Rightarrow x<0 \end{gathered}`

Since by hypothesis:

`x+5\ge0`

then also:

`-5\le x`

Then, for case 1, we have that:

`-5\le x<0`

Case 2: x+5 is lower than 0

In this case, |x+5|=-x-5, then:

`-x-5<5`

add x-5 to both sides:

[tex]\begin{gathered} -x-5+x-5<5+x-5 \\ \Rightarrow-10Since x+5 is lower than 0, then: