Question

solve for x and graph the solution.

$x^{2}-9x + 20 \leq 0$

plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.

submit

Explanation:

Step1: Factor the quadratic expression

$$x^{2}-9x + 20=(x - 4)(x - 5)$$
So the inequality becomes $(x - 4)(x - 5)\leq0$.

Step2: Find the roots of the corresponding equation

Set $(x - 4)(x - 5)=0$. Using the zero - product property $a\times b = 0$ implies $a = 0$ or $b = 0$.
If $x-4=0$, then $x = 4$; if $x - 5=0$, then $x=5$.

Step3: Test intervals

We have three intervals to test: $(-\infty,4)$, $(4,5)$ and $(5,\infty)$.

• For the interval $(-\infty,4)$, let $x = 3$. Then $(3 - 4)(3 - 5)=(-1)\times(-2)=2>0$.
• For the interval $(4,5)$, let $x = 4.5$. Then $(4.5 - 4)(4.5 - 5)=(0.5)\times(-0.5)=-0.25<0$.
• For the interval $(5,\infty)$, let $x = 6$. Then $(6 - 4)(6 - 5)=2\times1 = 2>0$.

Since the inequality is $(x - 4)(x - 5)\leq0$, the solution of the inequality includes the interval where the expression is non - positive and the roots.

Answer:

The solution of the inequality $x^{2}-9x + 20\leq0$ is $4\leq x\leq5$. On the number line, we plot closed circles at $x = 4$ and $x = 5$ and shade the segment between them.