Solve the inequality. Write your answer using interval notation. 2 ≤|x^2-9|<9

Name: Problem 27, Chapter 2: Precalculus
Uploaded: 2021-09-08T18:13:41-08:00
Duration: 6 min 59 s
Channel: Teresa Fuston
Description: Solve the inequality. Write your answer using interval notation. 2 ≤|x^2-9|<9

Solve the inequality. Write your answer using interval...

Key Concepts

Absolute Value Inequality

An absolute value inequality involves an expression inside absolute value symbols being bounded by constants. To solve such inequalities, one must consider the definition of absolute value, which means expressing the inequality in terms of cases where the inner expression is non-negative and where it is negative.

Case Analysis

Case analysis is the process of breaking down the inequality into separate conditions based on the sign of the expression inside the absolute value. This is crucial because the absolute value function is defined piecewise, leading to different resulting inequalities for the positive and negative cases.

Quadratic Equations and Inequalities

Quadratic equations and inequalities involve expressions where the variable is squared. When such expressions appear inside an absolute value, solving the inequality necessitates analyzing the quadratic expression, factoring or using the quadratic formula, and determining the intervals where the inequality holds.

Interval Notation

Interval notation is a concise way to express a range of values for which an inequality holds true. It uses symbols such as brackets and parentheses to indicate whether endpoints are included or excluded, providing a clear and standardized representation of the solution set.

Transcript

00:01 For this question, we are going to solve a quadratic compound inequality.

00:05 I'm going to write the compound inequality as two individual equations.

00:14 So i'm going to say 2 is equal to absolute value of x squared minus 9, and absolute value of x squared minus 9 is equal to 9.

00:29 Then from each i can write two equations.

00:35 X squared minus 9 is equal to positive 2 and x squared minus 9 is equal to negative 2.

00:45 For the other equation, x squared minus 9 is equal to positive 9 and x squared minus 9 is equal to negative 9.

00:58 Let's solve each equation.

01:00 First equation, add 9 and take the square root.

01:09 Second equation add 9 and take the square root third equation add 9 take the square root last equation add 9 and take the square root we have a bunch of numbers to put on our number line and they need to go in numeric order 0 will be in the middle and i'm not going to worry about spacing so much let's see square root of 8 3 root 2, the negative version would be farthest away, then the negative root 11, then the negative root 7, positive root 7, positive root 11, positive root 18, or 3 root 2.

02:21 And we need to test intervals.

02:25 We're looking for where x squared absolute value of x squared minus 9 is greater than or equal to 2 and less than 9.

02:45 So let's try, oh, let's see, negative 3 root 2 has a decimal value of negative 4.

02:55 So let's try negative 5.

03:01 In the middle, negative 5 squared is 25, 25, 25 minus 5.

03:05 Minus 9 is 16.

03:08 That is bigger than 2, but not less than 9.

03:14 Square root of 11 is 3 .3.

03:21 And let's try negative 4.

03:28 Negative 4 squared is 16.

03:31 16 minus 9 is between 2 and 9.

03:36 We will shade there.