QuestionFind the set $S=\{x:|x^{2}-6|>7\}$ as a union of intervals.

Studdy Solution

STEP 1

Assumptions1. The set is defined as $=\left\{x:\left|x^{}-6\right|>7\right\}$ . . We need to describe this set as a union of finite or infinite intervals.

STEP 2

The absolute value inequality $|x^{2}-6|>7$ can be split into two separate inequalities $x^{2}-6>7$ and $x^{2}-6<-7$ .

STEP 3

First, let's solve the inequality $x^{2}-6>7$ .

STEP 4

To solve the inequality $x^{2}-6>7$ , we first move the7 to the left side of the inequality to isolate $x^{2}$ .
$x^{2}-6-7>0$

STEP 5

implify the inequality to get $x^{2}>13$ .

STEP 6

To solve for x, we take the square root of both sides. Remember that when we take the square root of both sides of an inequality, we get two solutions one positive and one negative.
$x>\sqrt{13} \quad \text{and} \quad x<-\sqrt{13}$

STEP 7

Now, let's solve the second inequality $x^{2}-6<-7$ .

STEP 8

To solve the inequality $x^{2}-6<-7$ , we first move the7 to the left side of the inequality to isolate $x^{2}$ .
$x^{2}-6+7<0$

STEP 9

implify the inequality to get $x^{2}<$ .

STEP 10

To solve for x, we take the square root of both sides. Remember that when we take the square root of both sides of an inequality, we get two solutions one positive and one negative.
$x<\sqrt{} \quad \text{and} \quad x>-\sqrt{}$

STEP 11

However, the inequality $x^{}-6<-7$ has no solution because $x^{}-6$ is always greater than or equal to $-6$ , never less than $-7$ .

STEP 12

So, the solution to the original inequality $|x^{2}-6|>7$ is the union of the solutions to the two inequalities $x^{2}-6>7$ and $x^{2}-6<-7$ .

STEP 13

The solution to the inequality $|x^{2}-6|>7$ is the union of the intervals $(-\infty, -\sqrt{13})$ and $(\sqrt{13}, \infty)$ .
Therefore, the set $=\left\{x:\left|x^{2}-6\right|>7\right\}$ can be described as the union of the intervals $(-\infty, -\sqrt{13})$ and $(\sqrt{13}, \infty)$ .

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