Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |-2 x|>|-3| $$

Short answer

The solution is \((-\infty, -\frac{3}{2}) \cup (\frac{3}{2}, \infty)\).

Step-by-step solution

- Simplify the inequality

Start by evaluating the absolute value expressions. Remember that the absolute value of a number is always positive or zero. Therefore, \(|-2x| = 2|x|\) because the absolute value of \(-2x\) is the same as \(|2x|\). Similarly, \(|-3| = 3\). The inequality simplifies to \(|2x| > 3\).

- Split into two inequalities

To solve \(|2x| > 3\), consider the two scenarios for the absolute value: \(2x > 3\) and \(2x < -3\).

- Solve for \(x\) from \(2x > 3\)

Divide both sides by 2 to isolate \(x\). \[ 2x > 3 \implies x > \frac{3}{2} \]

- Solve for \(x\) from \(2x < -3\)

Divide both sides by 2 to isolate \(x\). \[ 2x < -3 \implies x < -\frac{3}{2} \]

- Combine solutions

The solution to the inequality \(|2x| > 3\) is \(x > \frac{3}{2} \) or \(x < -\frac{3}{2} \). In interval notation, this is \[(-\infty, -\frac{3}{2}) \cup (\frac{3}{2}, \infty)\].

- Graph the solution set

On a number line, draw open circles at \(-\frac{3}{2} \text{ and }\ \frac{3}{2}\). Shade everything to the left of \(-\frac{3}{2}\) and to the right of \(\frac{3}{2}\) to represent the solutions.

Key concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value signs, like \(|2x|>3\). The absolute value of a number represents its distance from zero on the number line, regardless of direction. It’s always non-negative. To solve absolute value inequalities, you need to consider both the positive and negative scenarios. For example:
\[|2x| > 3\] means that either:
\[2x > 3 \text{ or } 2x < -3\]
After setting up these two inequalities, solve each one separately to find the complete solution.

Interval Notation

Interval notation is a shorthand way to write solutions to inequalities. It uses brackets and parentheses to describe which numbers are included in the set. For instance:
\[x > \frac{3}{2} \text{ or } x < -\frac{3}{2}\]
In interval notation, this would be written as:
\[(-\frac{3}{2}, \frac{3}{2}) \text{ or, more correctly, } (-\frac{3}{2}, \frac{3}{2})\]
Parentheses mean that the endpoints are not included, which matches our solution where the inequality symbols are strict (>) and (<).

Graphing Inequalities

Graphing inequalities involves representing the solution set on a number line. For the inequality solution:
\[x > \frac{3}{2} \text{ or } x < -\frac{3}{2}\]
You would draw open circles at \(-\frac{3}{2}\) and \( \frac{3}{2}\) to indicate that these points are not included. Then, shade the regions to the left of \(-\frac{3}{2}\) and to the right of \( \frac{3}{2}\). This shaded area shows all the possible values of x that satisfy the inequality.

Set Notation

Set notation is another way to describe the solution sets of inequalities. It involves using curly braces and typically a format like this:
\[ \{ x \text{ | condition} \}\]
In our example, the condition is that x must be either greater than \( \frac{3}{2} \) or less than \(-\frac{3}{2}\). So, we write:
\[ \{ x \text{ | } x > \frac{3}{2} \text{ or } x < -\frac{3}{2} \}\]
This notation emphasizes the values of x that satisfy the inequality, focusing on inclusiveness without bounds.