Q12. - Algebra 1

Question

Solve $\left|x-4\right|<2$

Step-by-Step Solution

Answer

The solution for the given inequality $\left|x-4\right|<2$ is $\mathit{x}\mathbf{\in }\left(2,6\right)$

Step1. Given

$\left|x-4\right|<2$

Write the solution for the inequality | x | &#60; a

The solution to the inequality $\left|x\right|<a$ is $x<a$$x>-a$and $x<a$

That implies the solution of the inequality $\left|x\right|<a$ is the intersection of the solutions of the inequalities $x>-a$ and $x<a$.

Step3. Solve the given inequality x &#8722; 4 &#60; 2 .

The solution to the given inequality $\left|x-4\right|<2$ is

Case I: $x-4$ is non-negative.

$\begin{array}{c}x-4<2\\ x-4+4<2+4\\ x<6\\ x\in \left(-\infty ,6\right)\end{array}$

Case 2: $x-4$ is negative.

$\begin{array}{c}-\left(x-4\right)<2\\ \left(-1\right)\left(-\left(x-4\right)\right)>\left(-1\right)\left(2\right)\\ x-4>-2\\ x-4+4>-2+4\\ x>2\\ x\in \left(2,\infty \right)\end{array}$

The solution to the inequality$\left|x\right|<a$ is $x>-a$ and $x<a$

That implies the solution of the inequality $\left|x\right|<a$ is the intersection of the solutions of the inequalities $x>-a$ and $x<a$.

Find the intersection of the solutions of the inequalities $x-4<2$ and $x-4>-2$ find the solution of the inequality  $\left|x-4\right|<2$.

The intersection of the solutions of the inequalities $x-4<2$ and $x-4>-2$ is : $x\in \left(2,6\right)$

Therefore, the solution to the inequality $\left|x-4\right|<2$ is $x\in \left(2,6\right)$.