Chapter 6.3, Problem 50SR

To determine

Which inequality has the same solution set as $|2x-12|\ge 6$ ?

Answer to Problem 50SR

B. $|x-6|\ge 3$

Explanation of Solution

Given information: The inequalities are,

  $|2x-12|\ge \mathrm{6............................}(1)$

  $|12-2x|\le \mathrm{6............................}(A)$

  $|x-6|\ge \mathrm{3............................}(B)$

  $|12-2x|\ge -\mathrm{6............................}(C)$

  $|x-6|\ge \mathrm{9............................}(D)$

Calculation:

Solving (1),

  $\begin{array}{l}|2x-12|\ge 6\\ \Rightarrow -6\ge 2x-12\ge 6\\ \Rightarrow -6+12\ge 2x-12+12\ge 6+12\end{array}$

  $\begin{array}{l}\Rightarrow 6\ge 2x\ge 18\\ \Rightarrow \frac{6}{2}\ge \frac{2x}{2}\ge \frac{18}{2}\\ \Rightarrow 3\ge x\ge 9\end{array}$

Solving (A),

  $\begin{array}{l}|12-2x|\le 6\\ \Rightarrow -6\le 12-2x\le 6\\ \Rightarrow -6-12\le 12-2x-12\le 6-12\end{array}$

  $\begin{array}{l}\Rightarrow -18\le -2x\le -6\\ \Rightarrow 18\ge 2x\ge 6\\ \Rightarrow \frac{18}{2}\ge \frac{2x}{2}\ge \frac{6}{2}\\ \Rightarrow 9\ge x\ge 3\end{array}$

Solving (B),

  $\begin{array}{l}|x-6|\ge 3\\ \Rightarrow -3\ge x-6\ge 3\\ \Rightarrow -3+6\ge x-6+6\ge 3+6\\ \Rightarrow 3\ge x\ge 9\end{array}$

Solving (C),

  $\begin{array}{l}|12-2x|\ge -6\\ \Rightarrow 6\ge 12-2x\ge -6\\ \Rightarrow 6-12\ge 12-2x-12\ge -6-12\end{array}$

  $\begin{array}{l}\Rightarrow -6\ge -2x\ge -18\\ \Rightarrow 6\le 2x\le 18\\ \Rightarrow \frac{6}{2}\le \frac{2x}{2}\le \frac{18}{2}\\ \Rightarrow 3\le x\le 9\end{array}$

Solving (D),

  $\begin{array}{l}|x-6|\ge 9\\ \Rightarrow -9\ge x-6\ge 9\\ \Rightarrow -9+6\ge x-6+6\ge 9+6\\ \Rightarrow -3\ge x\ge 15\end{array}$

From the solutions it is observed that solutions of (1) and (B) are equivalent.

Thus, $|x-6|\ge 3$ has the same solution set as $|2x-12|\ge 6$ .