Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

Solve $x^2-10x+24\ge 0$

Step-by-Step Solution

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 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 $(-\infty ,4] \mathrm{or} [6,\infty )$.