Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(0<(2 x-4)^{-1}<\frac{1}{2}\)

Short answer

(2, \frac{9}{4})

Step-by-step solution

Simplify the inequality

Consider the inequality: \(0 < (2x - 4)^{-1} < \frac{1}{2}\)

Remove the inverse

Rewrite the inequality by expressing the inverted term: \(0 < \frac{1}{2x - 4} < \frac{1}{2}\)

Split into two inequalities

We split the compound inequality into two separate inequalities: 1) \(0 < \frac{1}{2x - 4}\) 2) \(\frac{1}{2x - 4} < \frac{1}{2}\)

Solve the first inequality

For the first inequality, \(0 < \frac{1}{2x - 4}\), we know that \(2x - 4 > 0\): \(2x - 4 > 0\)Solve for \(x\): \(2x > 4\)\(x > 2\)

Solve the second inequality

For the second inequality, \(\frac{1}{2x - 4} < \frac{1}{2}\): The inequality produces: \(2(2x - 4) > 1\)\(4x - 8 > 1\)Solve for \(x\): \(4x > 9\)\(x > \frac{9}{4}\)

Combine the solutions

Combine the two results: \(x > 2\) and \(x < 2\) implies \(2 < x < \frac{9}{4}\)

Express in interval notation

The interval notation for the solution is: \((2, \frac{9}{4})\)

Key concepts

Inequality Solutions

When faced with inequalities, we aim to find the range of values that make the inequality true. Unlike equations, inequalities show a range rather than a specific value.
To solve inequalities, we often:

• Isolate the variable on one side.
• Flip the inequality sign when multiplying or dividing by a negative number.
• Simplify complex expressions step-by-step.
• Check different ranges in the solution to ensure they satisfy the inequality.

For example, in solving \(0 < (2x - 4)^{-1} < \frac{1}{2}\), we transformed and split the inequalities to find a solution. Understanding how to manipulate and interpret these transformations is crucial for solving inequalities effectively.

Interval Notation

Interval notation is a shorthand way of writing inequalities. It uses brackets and parentheses to describe the set of solutions succinctly. Here are key symbols in interval notation:

• \((a, b)\) indicates all numbers between \(a\) and \(b\), excluding \(a\) and \(b\).
• \([a, b]\) includes both endpoints \(a\) and \(b\).
• \((a, b]\) or \([a, b)\) includes only one endpoint.
• \(-\infty\) and \(\infty\) denote unbounded intervals.

Using the interval notation makes it easy to convey complex solution sets. For our exercise, the solution \((2, \frac{9}{4})\) means \(x\) can be any value greater than 2 but less than \(\frac{9}{4}\). This notation neatly encapsulates the answer without listing all possible values.

Compound Inequalities

A compound inequality involves two separate inequalities combined into one statement by 'and' or 'or'. These are critical when a solution needs to satisfy multiple conditions.
In the exercise, we dealt with the compound inequality \(0 < (2x - 4)^{-1} < \frac{1}{2}\). It was broken into:

• \(0 < \frac{1}{2x-4}\)
• \(\frac{1}{2x-4} < \frac{1}{2}\)

Solving these separately, we found constraints for \(x\). The intersection of these solutions is the final answer. For example:

• For \(0 < \frac{1}{2x-4}\), we deduced \(x > 2\).
• For \(\frac{1}{2x-4} < \frac{1}{2}\), we found \(x < \frac{9}{4}\).

Combining these, we get \(2 < x < \frac{9}{4}\). Compound inequalities are powerful tools to find intersections of solutions, ensuring all conditions are met.