Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |2-3 x|>1 $$

Short answer

The solution in interval notation is \((-\infty, \frac{1}{3}) \cup (1, \infty)\).

Step-by-step solution

Understand the Absolute Value Inequality

We start with the inequality \(|2-3x| > 1\). Absolute value inequalities of the form \(|A| > B\) can be split into two separate inequalities: \(A > B\) or \(A < -B\).

Create Two Inequalities

Split \(|2 - 3x| > 1\) into two cases: \(2 - 3x > 1\) and \(2 - 3x < -1\).

Solve the First Inequality

Solve \(2 - 3x > 1\): \2 - 3x > 1\-3x > -1\text{ (subtract 2 from both sides)}\x < \frac{1}{3}\text{ (divide by -3 and reverse the inequality sign)}

Solve the Second Inequality

Solve \(2 - 3x < -1\): \2 - 3x < -1\-3x < -3\text{ (subtract 2 from both sides)}\x > 1\text{ (divide by -3 and reverse the inequality sign)}

Combine the Two Solutions

Combine the results from both inequalities: \(x < \frac{1}{3}\) or \(x > 1\).

Express in Interval Notation

In interval notation, the solution is \((-\infty, \frac{1}{3}) \cup (1, \infty)\).

Graph the Solution Set

Graph the solution on the number line: open circles at \(\frac{1}{3}\) and 1, with shading to the left of \(\frac{1}{3}\) and to the right of 1.

Key concepts

inequality solving

In this example, we're solving the absolute value inequality \(|2-3x|>1\). When dealing with absolute value inequalities of the form \(|A| > B\), we split it into two separate inequalities: \(A > B\) or \( A < -B\). This approach helps us get rid of the absolute value sign and makes the problem more manageable. Here, we split \(|2-3x|>1\) into two inequalities: \(2 - 3x > 1\) and \(2 - 3x < -1\). Then, we solve each inequality one by one. First, for \(2 - 3x > 1\), subtract 2 from both sides to get \(-3x > -1\). Divide both sides by -3, remembering to reverse the inequality sign, so \(x < \frac{1}{3}\). Similarly, for \(2 - 3x < -1\), subtract 2 from both sides to get \(-3x < -3\). Then divide by -3 and reverse the inequality sign to get \(x > 1\). Combining these, our solution is \(x < \frac{1}{3}\) or \(x > 1\).

interval notation

Interval notation is a concise way of writing subsets of the real number line. We use it to express the solution sets of inequalities. The solution from our previous example, \(x < \frac{1}{3}\) and \(x > 1\), can be written in interval notation as \((-\rightarrow, \frac{1}{3}) \bigcup (1, \rightarrow)\). This means we're including all values less than \( \frac{1}{3}\) and all values greater than 1. In interval notation:

• \(-\rightarrow\) represents all values less than a certain number extending to negative infinity.
• \(\rightarrow \) represents all values greater than a certain number extending to positive infinity.

The union symbol \(\bigcup \) indicates that we combine the two intervals. So, we're stating that our solution includes any number less than \( \frac{1}{3} \) alongside any number greater than 1.

inequality graphing

To visualize the solution, we graph it on a number line. This helps us see the ranges of values that satisfy the inequality. For our solution, \(x < \frac{1}{3}\) or \(x > 1\), we'll show this using the following steps:

• Draw a number line.
• Mark open circles at \( \frac{1}{3} \) and 1, as these points are not included in our solution.
• Shade the regions to the left of \( \frac{1}{3} \) and to the right of 1.

Open circles indicate that the endpoints themselves are not part of the solution, and shading shows the direction where the inequality holds.

absolute value properties

Absolute value inequalities use properties of absolute values to break down complex expressions. Recall that the absolute value \(|A|\) represents the distance of \( A \) from zero on a number line. This distance can never be negative. For an absolute value inequality like \(|2-3x| > 1\), we're looking for all \( x \) values where the distance between \(2-3x\) and 0 is greater than 1. This results in forming two separate inequalities because we consider both the positive and negative sides:

• \(A > B\) covers the scenario where \(A\) is positive and greater than \(B\).
• \(A < -B\) covers the scenario where \(A\) is negative and less than \(-B\).

By understanding these properties, splitting the inequality makes it easier to solve them.