Question

Solve the inequality $x^2-5x\le 24$. Graph the solution set.

Short answer

The solution set of given inequality is $[-3,8]$and its graph is

Step-by-step solution

Step 1. Using mathematical operations, solve the inequality.

First replace inequality symbol with an equality symbol.

$$\begin{gathered} x^2-5x=24 \\ {x}^2-5x-24=0 \end{gathered}$$

Now factorize the quadratic equation.

Find two terms whose sum is -5x and multiplication $-24x^2$.

$$\begin{gathered} x^2-8x+3x-24=0 \\ x(x-8)+3(x-8)=0 \end{gathered}$$

Taking common terms out.

$(x-8)(x+3)=0$gives

$$\begin{gathered} x-8=0orx+3=0 \\ x=8orx=-3 \end{gathered}$$

As the inequality was not strict in question, so the solution set of$x^2-5x\le 24$will be$-3\le x\le 8$or$[-3,8]$.

Step 2. The graph to show the solution set of x2-5x≤24 is as follows.