Question

NO LINKS!! Please help me with this problem. Part 5gg​

Answer 1

(-4, -1]

Explanation:

Given inequality:

`(x+10)/(x+4) \geq 3`

When the denominator of a rational function is zero, the function is undefined. Therefore, x ≠ -4, so x = -4 is a boundary point for the solution to the inequality.

Solve the inequality as though it were an equation.

`\implies(x+10)/(x+4) =3`

`\implies x+10 =3(x+4)`

`\implies x+10 =3x+12`

`\implies -2x=2`

`\implies x=-1`

The real solution to the equation is also a boundary point for the solution to the inequality.

Make the x = -1 boundary point a closed circle as the original inequality includes equality.

Make the x = -4 boundary point an open circle as it is not included.

Three regions have been created:

• x < -4
• -4 > x ≤ -1
• x ≥ -1

Select points from the different regions and test to see if they satisfy the original inequality:

`x=-5 \implies (-5+10)/(-5+4)=(5)/(-1)=-5 \\geq 3`

`x=-2 \implies (-2+10)/(-2+4)=(8)/(2)=4 \geq 3`

`x=0 \implies (0+10)/(0+4)=(10)/(4)=2.5 \\geq 3`

Since x = -5 does not satisfy the original inequality, the region x < -4 is not part of the solution.

Since x = -2 satisfies the original inequality, the region -4 > x ≤ -1 is part of the solution.

Since x = 0 does not satisfy the original inequality, the region x ≥ -1 is not part of the solution.

Therefore, the solution in interval notation is:

• (-4, -1]

To graph the solution set:

• Place an open circle at x = -4.
• Place a closed circle at x = -1.
• Connect the circles with a line between them.