Question

Solve$x^2-10x+24\ge 0$

Short answer

The solution of inequality is$\left\{\overline{)x}x\le 4\mathrm{or}x\ge 6\right\}$or in interval notation $(-\infty ,4]\mathrm{or}[6,\infty )$.

Step-by-step solution

Step 1. Given information. 

Solve the given quadratic inequality.

$x^2-10x+24\ge 0$

Step 2. Graph the function.

Graph the function $f\left(x\right)=x^2-10x+24$.

Step 3. Find intercept.

y- intercept:$f\left(0\right)=0^2-10\left(0\right)+24=24$

x- intercept: Solve $f\left(x\right)=0$

$$\begin{gathered} x^2-10x+24=0 \\ {x}^2-4x-6x+24=0 \\ x(x-4)-6(x-4)=0 \\ (x-4)(x-6)=0 \\ x=4,x=6 \end{gathered}$$

So, the x-intercept are $x_1=4\mathrm{and}{x}_2=6$.

The solution set is$\left\{\overline{)x}x\le 4\mathrm{or}x\ge 6\right\}$and in interval notation $(-\infty ,4]\mathrm{or}[6,\infty )$.

Step 4. Conclusion 

The solution set of inequality is $\left\{\overline{)x}x\le 4\mathrm{or}x\ge 6\right\}$or in interval notation$(-\infty ,4]\mathrm{or}[6,\infty )$.