Chapter 2, Problem 11CT

To determine

To solve: the absolute inequality $-2\left|y-3\right|-5\ge -4$ and graph the solution.

Answer to Problem 11CT

No solution

Explanation of Solution

Calculation:

Consider the absolute inequality,

  $-2\left|y-3\right|-5\ge -4$

Add 5 to both sides of the inequality,

  $\begin{array}{c}-2\left|y-3\right|-5+5\ge -4+5\\ -2\left|y-3\right|\ge 1\end{array}$

Divide both sides of the inequality by -2 (since the inequality is being divided by a negative number so the signs of the inequality will be reversed),

  $\begin{array}{c}\frac{-2\left|y-3\right|}{-2}\le \frac{1}{-2}\\ \left|y-3\right|\le -\frac{1}{2}\end{array}$

The right hand side of this inequality is a negative number whereas the left hand side of this inequality is absolute value of a number of it will always be positive.

And here the inequality symbol used is $\le$ , but it is impossible for a positive number to be less than or equal to a negative number.

Hence, the given inequality has no solution.