Question

1. solve and graph the inequality 36x² - 144 < 0

Explanation:

Step1: Rewrite as an equation and factor

Set $36x^{2}-144 = 0$. Factor out the greatest - common factor first. $36(x^{2}-4)=0$. Then use the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, so $36(x + 2)(x - 2)=0$.

Step2: Solve the equation for x

Set each factor equal to zero:
If $x+2 = 0$, then $x=-2$; if $x - 2=0$, then $x = 2$.

Step3: Test intervals

The intervals to test are $x\lt - 2$, $-2\lt x\lt2$, and $x\gt2$.
For $x\lt - 2$, let $x=-3$. Then $36(-3)^{2}-144=36\times9 - 144=324-144 = 180\gt0$.
For $-2\lt x\lt2$, let $x = 0$. Then $36(0)^{2}-144=-144\lt0$.
For $x\gt2$, let $x = 3$. Then $36(3)^{2}-144=36\times9-144 = 180\gt0$.

Answer:

The solution of the inequality $36x^{2}-144\lt0$ is $-2\lt x\lt2$. On the number - line, we use open circles at $x=-2$ and $x = 2$ and shade the region between them.