Question

solve the inequality. express your answer using interval notation. graph the solution set. (x + 6)(x - 7)>(x - 3)(x + 3) the solution to the inequality is (type your answer in interval notation. use integers or fractions for any numbers in the expression. simplify your answer.) graph the inequality. choose the correct graph below.

Explanation:

Step1: Find the roots

Set each factor equal to zero: $x + 6=0$ gives $x=-6$, $x - 7=0$ gives $x = 7$, $x-3=0$ gives $x = 3$, $x + 3=0$ gives $x=-3$.

Step2: Test intervals

The intervals are $(-\infty,-6)$, $(-6,-3)$, $(-3,3)$, $(3,7)$ and $(7,\infty)$.
For $x<-6$, let $x=-7$, then $(-7 + 6)(-7-7)(-7 - 3)(-7+3)=(-1)\times(-14)\times(-10)\times(-4)=560>0$.
For $-6For $-30$.
For $3For $x>7$, let $x = 8$, then $(8 + 6)(8-7)(8 - 3)(8+3)=(14)\times(1)\times(5)\times(11)=770>0$.

Step3: Write the solution in interval notation

The solution of the inequality $(x + 6)(x - 7)(x - 3)(x+3)>0$ is $(-\infty,-6)\cup(-3,3)\cup(7,\infty)$.

Step4: Graph the solution

The graph will have open - circles at $x=-6,x=-3,x = 3,x = 7$ and will be shaded to the left of $-6$, between $-3$ and $3$ and to the right of $7$.

Answer:

The solution in interval notation is $(-\infty,-6)\cup(-3,3)\cup(7,\infty)$. The correct graph is one that has open - circles at $x=-6,x=-3,x = 3,x = 7$ and is shaded to the left of $-6$, between $-3$ and $3$ and to the right of $7$ (description of the correct graph based on the options not shown completely here but the above is the correct solution details).