Chapter SH, Problem 4.24EP

To determine

Find the solution of the given inequality.

Answer to Problem 4.24EP

The solutions set is $-4\le x\le 7$ or $\left[-4,7\right]$ .

Explanation of Solution

Given:

The given inequalityis $x^2-3x\le 28$ .

Calculation:

Related quadratic equation is

  $x^2-3x=28$

Subtract $28$ from both sides.

  $\begin{array}{l}\Rightarrow x^2-3x-28=28-28\\ \Rightarrow x^2-3x-28=0\\ \Rightarrow x^2-7x+4x-28=0\\ \Rightarrow x\left(x-7\right)+4\left(x-7\right)=0\\ \Rightarrow \left(x-7\right)\left(x+4\right)=0\\ \Rightarrow x=-4,7\end{array}$

Plot $-4$ and $7$ on number line.

  

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

Test $x\le -4$ .

  $\begin{array}{l}{x}^2-3x\le 28\\ \Rightarrow {\left(-6\right)}^2-3\left(-6\right)\le 28\\ \Rightarrow 36+18\le 28\\ \Rightarrow 54<\ne 28\end{array}$

Test $-4\le x\le 7$

  $\begin{array}{l}{x}^2-3x\le 28\\ \Rightarrow 0^2-3\cdot 0\le 28\\ \Rightarrow 0\le 28\end{array}$

Test $x\ge 7$

  $\begin{array}{l}{x}^2-3x\le 28\\ \Rightarrow {\left(8\right)}^2-3\left(8\right)\le 28\\ \Rightarrow 64-24\le 28\\ \Rightarrow 40<\ne 28\end{array}$

Hence thesolutions set is $-4\le x\le 7$ or $\left[-4,7\right]$ .