Chapter 5.3, Problem 45PPS

To determine

To solve the inequality:

  $\frac{2x-4}{6}\ge -5x+2$

And check the solution.

Answer to Problem 45PPS

  $[\frac{1}{2},\infty )$

Explanation of Solution

Given:

  $\frac{2x-4}{6}\ge -5x+2$

Concept Used:.

• To check the answer solution for the given inequality

1. First check the solution at end point by converting it into an equation.
2. If the solution is $x<m$, then pick any value of x less than m and check whether it satisfies the inequality.
3. If the solution is $x>m$, then pick any value of x greater than m and check whether it satisfies the inequality.

Properties of inequality:

1. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

To solve the inequality:

  $\frac{2x-4}{6}\ge -5x+2$

First multiply both sides of the inequality by $6$ , as

  $\begin{array}{l}\frac{2x-4}{6}\ge -5x+2\\ \Rightarrow 6\times \frac{2x-4}{6}\ge 6\left(-5x+2\right)\\ \Rightarrow 2x-4\ge \left(-30x+12\right)\\ \Rightarrow 2x+30x-4\ge \left(-30x+30x+12\right),\text{ adding 30}x\text{ on bothsides}\text{.}\\ \Rightarrow 32x-4\ge 12\\ \Rightarrow 32x-4+4\ge 12+4,\text{ Adding 4 on bothsides}\text{.}\\ \Rightarrow 32x\ge 16\\ \Rightarrow \frac{32x}{32}\ge \frac{16}{32},\text{ Dividing bothsides by 32}\\ \Rightarrow x\ge \frac{1}{2}\end{array}$

Thus, the inequality is true for x greater than or equal to $\frac{1}{2}$ , thus the solution of given inequality in interval form is:

  $[\frac{1}{2},\infty )$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting x equals to $\frac{1}{2}$ on both sides as,

  $\begin{array}{l}\frac{2x-4}{6}\ge -5x+2\\ \Rightarrow \frac{2\left(\frac{1}{2}\right)-4}{6}\ge -5\left(\frac{1}{2}\right)+2\\ \Rightarrow \frac{1-4}{6}\ge \frac{-5}{2}+2\\ \Rightarrow \frac{-1}{2}\ge -\frac{1}{2}\text{which is true}\text{.}\end{array}$

Now put any value any value of x greater than $\frac{1}{2}$ , say 1 and check whether it satisfies the inequality, as

  $\begin{array}{l}\frac{2x-4}{6}\ge -5x+2\\ \Rightarrow \frac{2\left(1\right)-4}{6}\ge -5\left(1\right)+2\\ \Rightarrow \frac{-2}{6}\ge -3\\ \Rightarrow \frac{-1}{3}\ge -3\\ \text{ Which is true}\text{.}\end{array}$

Since, both conditions are satisfied thus solution of inequality is correct