Question

salgtrig5 1.8.042. solve the nonlinear inequality. express the solution using interval notation. $x^{2}

Explanation:

Step1: Rearrange the inequality

Move all terms to one - side: $x^{2}-x - 30<0$.

Step2: Factor the quadratic expression

Factor $x^{2}-x - 30$ as $(x - 6)(x+5)<0$.

Step3: Find the roots

Set $(x - 6)(x + 5)=0$. The roots are $x=6$ and $x=-5$.

Step4: Test intervals

Test the intervals $(-\infty,-5)$, $(-5,6)$ and $(6,\infty)$.
For $x<-5$, let $x=-6$, then $(-6 - 6)(-6 + 5)=(-12)\times(-1)=12>0$.
For $-5For $x>6$, let $x = 7$, then $(7 - 6)(7 + 5)=1\times12=12>0$.

Answer:

The solution in interval notation is $(-5,6)$.
To graph the solution set on the number - line:

1. Place open circles at $x=-5$ and $x = 6$ (because the inequality is strict, i.e., $<$ not $\leq$).
2. Shade the region between $-5$ and $6$.