SOLUTION: Solve the inequality x(x + 6) > 16 - x^2 + 14x. Write your answer in interval notation.

Question 1209290: Solve the inequality x(x + 6) > 16 - x^2 + 14x. Write your answer in interval notation.
Found 2 solutions by mccravyedwin, math_tutor2020:
Answer by mccravyedwin(408) (Show Source): You can put this solution on YOUR website!

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Consider the equation

Let's get everything to one side.

That last equation is of the format
where,
a = 1, b = -4, c = -8
Plug those into the quadratic formula.

or

or both of which are approximate.

--------------------------------------------------------------------------

Draw a number line with regions A, B, C
region A = stuff to the left of -1.464
region B = stuff between -1.464 and 5.464
region C = stuff to the right of 5.464

Possible test values for regions A,B,C could be: x = -2, x = 0, x = 6 in that order
You plug each test value back into the original inequality to see if we get a true statement or not.

If we tried x = -2, then,

which is true.
Any x value in region A is in the solution set for

Region A is aka
Yielding the interval notation
Use curved parenthesis to indicate we do not include the endpoints in the solution set.

If we tried x = 0 then

which is false.
Any x value in region B, ie the region between and will make the original inequality false.

The last region to test is region C.
Let's plug in x = 6

which is true
Therefore aka translates to the interval notation

We found these two interval notation regions
or

Glue them together with a set union symbol to end up with this final answer

You can use a graphing calculator like Desmos and GeoGebra to confirm.

RELATED QUESTIONS
Solve the inequality. Write your answer in interval notation. -12<5-x<16... (answered by josgarithmetic)
Solve the inequality (x - 2)(x + 6) \le -x^2 + 5x - 15. Write your answer in interval... (answered by ikleyn)
Solve the following rational inequality. x>= 3/x+2 write your answer in interval... (answered by lwsshak3)
Solve the following inequality: (x-6)^2(x+9)< 0 (Type your answer in interval... (answered by lwsshak3)
Solve the inequality and write your answer in interval notation... (answered by jim_thompson5910)
Solve the inequality x^3 + 4x > 5x^2 - 20x + 8. Write your answer in interval... (answered by greenestamps)
Solve the inequality. Write your answer in interval notation. (x + 1)/((x + 5)(x -... (answered by Fombitz)
solve the inequality and write the answer in interval notation... (answered by stanbon)
solve the following inequality. write your solution in interval notation.... (answered by lwsshak3)