Problem 48 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ \frac{1}{2} x \leq 2 \text { and } 0.75 x \geq-6 $$

Step-by-Step Solution

Answer

The solution set is \([-8, 4]\) in interval notation.

Step 1: Solve the first inequality

The inequality is \( \frac{1}{2}x \leq 2 \). To isolate \( x \), multiply both sides by 2: \( x \leq 4 \).

Step 2: Solve the second inequality

The second inequality is \( 0.75x \geq -6 \). First, divide both sides by 0.75: \( x \geq \frac{-6}{0.75} \). Simplifying gives \( x \geq -8 \).

Step 3: Determine the solution set

Combine the solutions from both inequalities: \( -8 \leq x \leq 4 \). This is the overlapping region that satisfies both conditions.

Step 4: Write the interval notation

The solution set in interval notation is \([-8, 4]\).

Step 5: Graph the solution set

On a number line, draw a filled circle at \(-8\) and another at \(4\). Shade the region in between to represent all values \( x \) that satisfy \(-8 \leq x \leq 4\).

Key Concepts

Inequality Solving

When solving compound inequalities, the goal is to find a range of values that satisfy more than one inequality at the same time. Let's break down the approach with the example provided:

• First inequality: \( \frac{1}{2}x \leq 2 \). To isolate \(x\), multiply each side by 2, resulting in \(x \leq 4\).
• Second inequality: \(0.75x \geq -6 \). This involves dividing each side by 0.75 to isolate \(x\), yielding \(x \geq -8\).

The solution for a compound inequality often necessitates combining solutions from each separate inequality. Thus, the solution set that meets both conditions is \(-8 \leq x \leq 4\). Having a systematic approach by treating each part of the compound inequality separately aids in solving more complicated problems.

Interval Notation

Interval notation is a handy way to express the set of solutions for inequalities. It uses brackets and parentheses to describe the range of numbers that satisfy the inequality.

• Square brackets, like \([ ]\), indicate that an endpoint is included in the solution set.
• Parentheses, like \(( )\), signify that an endpoint is not part of the solution.

For example, the solution \(-8 \leq x \leq 4\) is written as \([-8, 4]\) in interval notation. Here, both endpoints are included, using square brackets because \(x\) can be equal to -8 and 4. Understanding this notation streamlines communication of an inequality's solution, offering a concise and consistent format that is crucial when working with more complex expressions.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of the solution set. To graph the inequality \(-8 \leq x \leq 4\), perform the following steps:

• Draw a number line with appropriate scale, ensuring that at least the endpoints -8 and 4 are clearly marked.
• Place filled circles (or dots) on the number line at \(-8\) and 4 to indicate these numbers are part of the solution set.
• Shade the region between the two numbers, showing all values \(x\) that satisfy the inequality.

Graphing assists in visualizing the solution, making it easier to interpret the range of values that meet the conditions of both inequalities. This graphic method is particularly useful for verifying the overlap of solution sets in compound inequalities.