Chapter SH, Problem 4.25EP

To determine

Find the solution of the given inequality.

Answer to Problem 4.25EP

The solutions set is $x\le 2.5$ and $x\ge 4$ or $\left(-\infty ,2.5\right]\cup \left[4,\infty \right)$ .

Explanation of Solution

Given:

The given inequalityis $2x^2-13x\ge -20$ .

Calculation:

Related quadratic equation is

  $2x^2-13x=-20$

Add $20$ to both sides.

  $\begin{array}{l}\Rightarrow 2x^2-13x+20=-20+20\\ \Rightarrow 2x^2-13x+20=0\\ \Rightarrow 2x^2-8x-5x+20=0\\ \Rightarrow 2x\left(x-4\right)-5\left(x-4\right)=0\\ \Rightarrow \left(x-4\right)\left(2x-5\right)=0\\ \Rightarrow x=\frac{5}{2},4\\ \Rightarrow x=2.5,4\end{array}$

Plot $2.5$ and $4$ on number line.

  

Test a value from each interval to see if it satisfies the original inequality.

Test $x\le 2.5$ .

  $\begin{array}{l}2x^2-13x\ge -20\\ \Rightarrow 2{\left(0\right)}^2-13\left(0\right)\ge -20\\ \Rightarrow 0\ge -20\end{array}$

Test $2.5\le x\le 4$

  $\begin{array}{l}2x^2-13x\ge -20\\ \Rightarrow 2{\left(3\right)}^2-13\left(3\right)\ge -20\\ \Rightarrow 18-39\ge -20\\ \Rightarrow -21\ne >-20\end{array}$

Test $x\ge 4$

  $\begin{array}{l}2x^2-13x\ge -20\\ \Rightarrow 2{\left(5\right)}^2-13\left(5\right)\ge -20\\ \Rightarrow 50-65\ge -20\\ \Rightarrow -15\ge -20\end{array}$

Hence thesolutions set is $x\le 2.5$ and $x\ge 4$ or $\left(-\infty ,2.5\right]\cup \left[4,\infty \right)$ .