Question

solve for x and graph the solution. -2x² + 14x - 24 < 0 plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.

Explanation:

Step1: Make the quadratic - inequality into standard form

First, multiply both sides of the inequality $-2x^{2}+14x - 24<0$ by - 1. Remember to reverse the inequality sign. We get $2x^{2}-14x + 24>0$. Then divide through by 2 to simplify: $x^{2}-7x + 12>0$.

Step2: Factor the quadratic expression

Factor $x^{2}-7x + 12$ as $(x - 3)(x - 4)>0$.

Step3: Find the roots of the corresponding quadratic equation

Set $(x - 3)(x - 4)=0$. The roots are $x = 3$ and $x = 4$.

Step4: Determine the solution intervals

We consider the three intervals: $(-\infty,3)$, $(3,4)$, and $(4,\infty)$.

• For $x<3$, let's take $x = 2$. Then $(2 - 3)(2 - 4)=(-1)\times(-2)=2>0$, so $(-\infty,3)$ is part of the solution.
• For $3
• For $x>4$, let's take $x = 5$. Then $(5 - 3)(5 - 4)=2\times1 = 2>0$, so $(4,\infty)$ is part of the solution.

Answer:

The solution of the inequality is $x<3$ or $x>4$. On the number - line, we have an open circle at $x = 3$ and an open circle at $x = 4$. The graph consists of two rays: one ray extends to the left of $x = 3$ and the other ray extends to the right of $x = 4$.