Math  /  Algebra

QuestionSolve the following inequality. x2+19x+90<0x^{2}+19 x+90<0
Select the correct choice below and, if necessary, fill in the answer box. A. The solution set is \square . (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. There is no real solution.

Studdy Solution

STEP 1

What is this asking? Find the range of xx values that make x2+19x+90x^2 + 19x + 90 less than zero. Watch out! Don't forget to consider the *less than* sign (<) and not just where the expression equals zero!

STEP 2

1. Factor the quadratic expression.
2. Find the roots of the quadratic equation.
3. Determine the sign of the quadratic expression in the intervals determined by the roots.

STEP 3

Let's **factor** this quadratic!
We're looking for two numbers that add up to **19** and multiply to **90**.
Think think think... **9** and **10**!
Perfect!

STEP 4

So, we can rewrite our inequality as (x+9)(x+10)<0(x+9)(x+10) < 0.
This factored form is much easier to work with!

STEP 5

To find where our expression is less than zero, it's super helpful to know where it *equals* zero first.
We set each factor equal to zero and solve.

STEP 6

For the first factor, x+9=0x+9=0, we subtract 9 from both sides (adding -9 to zero on the right side keeps it at zero), giving us x=9x = -9.

STEP 7

For the second factor, x+10=0x+10=0, we subtract 10 from both sides (adding -10 to zero on the right side keeps it at zero), giving us x=10x = -10.

STEP 8

So, our **roots** are x=10x = -10 and x=9x = -9.
These are the points where our parabola crosses the x-axis!

STEP 9

Now, let's **test** some values in each of the intervals created by our roots: x<10x < -10, 10<x<9-10 < x < -9, and x>9x > -9.

STEP 10

Let's pick x=11x = -11 for the first interval.
Plugging it into our factored inequality, we get (11+9)(11+10)=(2)(1)=2(-11+9)(-11+10) = (-2)(-1) = 2.
Since 2 is *greater* than 0, this interval doesn't satisfy our inequality.

STEP 11

Now, let's choose x=9.5x = -9.5 for the second interval.
Plugging it in, we have (9.5+9)(9.5+10)=(0.5)(0.5)=0.25(-9.5+9)(-9.5+10) = (-0.5)(0.5) = -0.25.
Since 0.25-0.25 is *less* than 0, this interval *does* satisfy our inequality!

STEP 12

Finally, let's try x=8x = -8 for the last interval.
We get (8+9)(8+10)=(1)(2)=2(-8+9)(-8+10) = (1)(2) = 2.
Since 2 is *greater* than 0, this interval doesn't work for us.

STEP 13

So, the only interval where x2+19x+90<0x^2 + 19x + 90 < 0 is between our two roots, 10-10 and 9-9.

STEP 14

The solution set is (10,9)(-10, -9).
So the answer is A.
The solution set is (-10,-9).

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