Question

question 1
solve the following inequality and graph the solution. answer using interval notation.
x^2 - 9 < 0
question help: video ebook written example
clear all draw:

Explanation:

Step1: Factor the left - hand side

Factor $x^{2}-9$ as $(x + 3)(x - 3)$. So the inequality becomes $(x + 3)(x - 3)<0$.

Step2: Find the roots

Set $(x + 3)(x - 3)=0$. The roots are $x=-3$ and $x = 3$. These roots divide the number line into three intervals: $(-\infty,-3)$, $(-3,3)$ and $(3,\infty)$.

Step3: Test the intervals

• For the interval $(-\infty,-3)$, let's choose $x=-4$. Then $( - 4 + 3)(-4 - 3)=(-1)\times(-7)=7>0$.
• For the interval $(-3,3)$, let's choose $x = 0$. Then $(0 + 3)(0 - 3)=(3)\times(-3)=-9<0$.
• For the interval $(3,\infty)$, let's choose $x = 4$. Then $(4 + 3)(4 - 3)=(7)\times(1)=7>0$.

Answer:

The solution of the inequality $x^{2}-9<0$ in interval notation is $(-3,3)$. To graph it, we draw an open - circle at $x=-3$ and $x = 3$ (because the inequality is strict, i.e., $<$ not $\leq$) and shade the region between them on the number line.