Question

graph the solution to the following inequality on the number line.
$x(x-7)\geq0$

Explanation:

Step1: Find critical points

Set $x(x-7)=0$, so $x=0$ or $x=7$.

Step2: Test intervals

Test $x<-1$: $(-1)(-8)=8>0$, valid.
Test $0Test $x>8$: $(8)(1)=8>0$, valid.

Step3: Include critical points

Inequality is $\geq0$, so $x=0$ and $x=7$ are included.

Step4: Graph on number line

• Closed dots at $0$ and $7$.
• Shade left of $0$ (including $0$) and right of $7$ (including $7$).

Answer:

The solution is $x\leq0$ or $x\geq7$. On the number line:

• Draw a closed dot at $0$ and shade all numbers to the left of $0$ (extending to negative infinity).
• Draw a closed dot at $7$ and shade all numbers to the right of $7$ (extending to positive infinity).