Question

12) Use the number line to solve the following inequality:

Answer 1

ANSWER and EXPLANATION

We are given the inequality:

$\mleft|\frac{x\text{ - 3}}{2}\mright|\text{ + 2 < 6}$

To solve it, we have to isolate the absolute value:

$\begin{gathered} \mleft|\frac{x\text{ - 3}}{2}\mright|\text{ < 6 - 2} \\ =>\text{ }\mleft|\frac{\text{x - 3}}{2}\mright|\text{ < 4} \end{gathered}$

Now, that absolute value means two things. It means that:

$\frac{x\text{ - 3}}{2}\text{ < 4 and }\frac{x\text{ - 3}}{2}\text{ > -4}$

Now, solve them separately:

$\begin{gathered} \Rightarrow\text{ x - 3 < 4 }\cdot\text{ 2 and x - 3 > -4 }\cdot\text{ 2} \\ x\text{ - 3 < 8 and x - 3 > -8} \\ \Rightarrow\text{ x < 8 + 3 and x > -8 + 3} \\ x\text{ < 11 and x > -5} \\ \Rightarrow\text{ -5 < x < 11} \end{gathered}$

Now represent that on the number line:

That is the number line.

Note: The circles are not shaded