Question Solve the inequality $x^{2} < x + 12$ and express the answer using interval notation.

Studdy Solution

STEP 1

Assumptions
1. We are solving the inequality $x^{2} < x + 12$ .
2. We will find the values of $x$ that satisfy the inequality.
3. We will express the solution using interval notation.

STEP 2

First, we need to bring all terms to one side of the inequality to set it to zero, which will allow us to identify the critical points.
$x^{2} - x - 12 < 0$

STEP 3

Next, we factor the quadratic expression on the left-hand side, if possible.
The factors of -12 that add up to -1 (the coefficient of $x$ ) are -4 and 3. So we can write:
$(x - 4)(x + 3) < 0$

STEP 4

Now, we find the critical points by setting each factor equal to zero.
For $x - 4 = 0$ , we get $x = 4$ .
For $x + 3 = 0$ , we get $x = -3$ .

STEP 5

We use the critical points to divide the number line into intervals and determine the sign of the expression $(x - 4)(x + 3)$ within each interval. The intervals are $(-\infty, -3)$ , $(-3, 4)$ , and $(4, \infty)$ .

STEP 6

Test a point from each interval in the inequality $(x - 4)(x + 3) < 0$ to determine if the expression is negative or positive in that interval.
For the interval $(-\infty, -3)$ , choose $x = -4$ . Plugging this into the expression gives $(x - 4)(x + 3) = (-4 - 4)(-4 + 3) = (-8)(-1) = 8$ , which is positive.

STEP 7

For the interval $(-3, 4)$ , choose $x = 0$ . Plugging this into the expression gives $(x - 4)(x + 3) = (0 - 4)(0 + 3) = (-4)(3) = -12$ , which is negative.

STEP 8

For the interval $(4, \infty)$ , choose $x = 5$ . Plugging this into the expression gives $(x - 4)(x + 3) = (5 - 4)(5 + 3) = (1)(8) = 8$ , which is positive.

STEP 9

Since we are looking for the intervals where the expression is negative, we only consider the interval where our test point gave us a negative result. This is the interval $(-3, 4)$ .

STEP 10

Express the solution using interval notation. The solution is the interval where the expression $(x - 4)(x + 3)$ is less than zero.
The solution in interval notation is:
$(-3, 4)$

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