Question

1. solve and graph the inequality x² - 23x - 108 > 0. change inequality to equal and factor. name factors of. solve for x. yes or no. graph answers on number line. check numbers in each region. solution and

Explanation:

Step1: Set the inequality as an equation and factor

Set $x^{2}-23x - 108=0$. We need to find two numbers that multiply to $- 108$ and add up to $-23$. The numbers are $-27$ and $4$. So, the factored form is $(x - 27)(x+4)=0$.

Step2: Solve for x

Using the zero - product property, if $(x - 27)(x + 4)=0$, then $x-27=0$ or $x + 4=0$. Solving these equations gives $x=27$ or $x=-4$.

Step3: Determine the solution intervals

We consider the three intervals determined by the roots $x=-4$ and $x = 27$: $(-\infty,-4)$, $(-4,27)$ and $(27,\infty)$.

• For the interval $(-\infty,-4)$, let's test $x=-5$. Then $(-5 - 27)(-5 + 4)=(-32)\times(-1)=32>0$, so this interval is part of the solution.
• For the interval $(-4,27)$, let's test $x = 0$. Then $(0 - 27)(0 + 4)=(-27)\times4=-108<0$, so this interval is not part of the solution.
• For the interval $(27,\infty)$, let's test $x=28$. Then $(28 - 27)(28 + 4)=1\times32 = 32>0$, so this interval is part of the solution.

Answer:

The solution of the inequality $x^{2}-23x - 108>0$ is $x<-4$ or $x>27$. In interval notation, the solution is $(-\infty,-4)\cup(27,\infty)$. To graph, we use open circles at $x=-4$ and $x = 27$ on the number - line and shade the regions to the left of $x=-4$ and to the right of $x = 27$.