# Solving an Inequality: |x-3| < 2|x|

• TheRedDevil18
In summary, the given inequality can be solved by using the formula |a| < |b| ↔ a^2 < b^2 and understanding that there are two possibilities for x: either x is greater than or equal to 3, or x is less than 3.
TheRedDevil18

## Homework Statement

Solve the given inequality by interpreting it as a statement about distances in the real line:

|x-3| < 2|x|

## The Attempt at a Solution

I have no clue what to do here and I do not understand the answer in the textbook

Goes something like this.....
x^2 - 6x + 9.....Have no idea how they got that
= (x-3)^2
...and so forth

There are two possibilities for x: either x is greater than or equal to 3, or x is less than 3.

If x ≥ 3, what does the inequality look like (i.e. without the absolute value)?

TheRedDevil18 said:

## Homework Statement

Solve the given inequality by interpreting it as a statement about distances in the real line:

|x-3| < 2|x|

## The Attempt at a Solution

I have no clue what to do here and I do not understand the answer in the textbook

Goes something like this.....
x^2 - 6x + 9.....Have no idea how they got that
= (x-3)^2
...and so forth

They used ##|a|<|b| \leftrightarrow a^2<b^2##.

## 1. What is an inequality?

An inequality is a mathematical statement that compares two quantities using symbols such as <, >, ≤, or ≥. It indicates that one quantity is less than, greater than, less than or equal to, or greater than or equal to the other.

## 2. What is the absolute value?

The absolute value of a number is its distance from 0 on the number line. It is always a positive value, and can be represented by two vertical lines surrounding the number, such as |3| = 3.

## 3. How do you solve an inequality with absolute values?

To solve an inequality with absolute values, you must first isolate the absolute value expression on one side of the inequality. Then, you can break the inequality into two separate inequalities, one with a positive value and one with a negative value. Finally, solve each inequality separately to find the range of values that satisfy the original inequality.

## 4. How do you solve an absolute value inequality with a variable on both sides?

To solve an absolute value inequality with a variable on both sides, you must first isolate the absolute value expression on one side of the inequality. Then, you can break the inequality into two separate inequalities, one with a positive value and one with a negative value. Finally, solve each inequality separately to find the range of values that satisfy the original inequality. This process may result in a union of two or more inequality solutions.

## 5. What is the solution to the inequality |x-3| < 2|x|?

The solution to the inequality |x-3| < 2|x| is any value of x that satisfies both of the inequalities |x-3| < 2x and |x-3| < -2x. This can be represented as a compound inequality, -2x < x-3 < 2x, which simplifies to -3 < x < 3. Therefore, the solution set for this inequality is all real numbers between -3 and 3, not including the endpoints.

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