Chapter 4.9, Problem 14WE

To determine

To find and graph the solutionset of inequality.

Answer to Problem 14WE

The solution set of given inequality is $\left\{x:x\ge \frac{1}{2}\text{ or }x\le \frac{-3}{2}\right\}$

Explanation of Solution

Given information:

The given inequality is $4x\left(x+1\right)\ge 3$

Calculation:

First write the inequality

  $\begin{array}{l}4x\left(x+1\right)\ge 3\\ 4x^2+4x-3\ge 0\left[\text{make one side 0}\right]\\ \left(2x-1\right)\left(2x+3\right)\ge 0\left[\text{factor of the polynomial}\right]\end{array}$

The product is non-negative if and only if both factors have same sign or at least one factor must be zero.

Case 1:

  $\begin{array}{l}2x-1\ge 0\text{ and }2x+3\ge 0\\ 2x\ge 1\text{ and }2x\ge -3\\ x\ge \frac{1}{2}\text{ and }x\ge \frac{-3}{2}\end{array}$

The solution set of this conjunction is $\left\{x:x\ge \frac{1}{2}\right\}$

Case 2:

  $\begin{array}{l}2x-1\le 0\text{ and }2x+3\le 0\\ 2x\le 1\text{ and }2x\le -3\\ x\le \frac{1}{2}\text{ and }x\le \frac{-3}{2}\end{array}$

The solution set of this conjunction is $\left\{x:x\le \frac{-3}{2}\right\}$

Therefore,

The solution set of given inequality is $\left\{x:x\ge \frac{1}{2}\text{ or }x\le \frac{-3}{2}\right\}$

The graph of the solution set is given below