Problem 70 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(-2 \leq \frac{5-3 x}{2} \leq 2\)

Step-by-Step Solution

Answer

The solution is \(\left[ \frac{1}{3}, 3 \right]\); graph as a line between \(\frac{1}{3}\) and \(3\) with solid dots.

Step 1: Understand the Compound Inequality

The compound inequality to solve is \(-2 \leq \frac{5-3x}{2} \leq 2\). This means we need to find all values of \(x\) that make both inequalities true simultaneously.

Step 2: Solve the First Inequality

Start by solving the inequality \(-2 \leq \frac{5-3x}{2}\). Multiply every term by 2 to get: \(-4 \leq 5 - 3x\). Then, solve for \(x\) by subtracting 5 from both sides: \(-9 \leq -3x\). Divide by -3 (remember to reverse the inequality sign): \(3 \geq x\) or \(x \leq 3\).

Step 3: Solve the Second Inequality

Now solve the inequality \(\frac{5-3x}{2} \leq 2\). Multiply every term by 2 to get: \(5 - 3x \leq 4\). Subtract 5 from both sides: \(-3x \leq -1\). Divide by -3 (reverse the sign): \(x \geq \frac{1}{3}\).

Step 4: Combine the Solutions

The solution to the compound inequality involves the intersection of \(x \leq 3\) and \(x \geq \frac{1}{3}\). This can be written as \(\frac{1}{3} \leq x \leq 3\).

Step 5: Write in Interval Notation

The solution \(\frac{1}{3} \leq x \leq 3\) is represented in interval notation as \(\left[ \frac{1}{3}, 3 \right]\).

Step 6: Graph the Solution Set

To graph \([\frac{1}{3}, 3]\), draw a number line. Place a solid dot at \(\frac{1}{3}\) and another at \(3\), shading the region in between to represent all numbers from \(\frac{1}{3}\) to \(3\), inclusive.

Key Concepts

Understanding Inequality Solving Steps

Compound inequalities may look complex at first glance, but tackling them step-by-step can make the process manageable. In our example, the compound inequality is \(-2 \leq \frac{5-3x}{2} \leq 2\). This requires ensuring that the variable \(x\) satisfies both smaller inequalities within the compound statement simultaneously.

The first step is separating the compound inequality into its two parts:

• \(-2 \leq \frac{5-3x}{2}\)
• \(\frac{5-3x}{2} \leq 2\)

Each part is solved using basic algebra, focusing on isolating \(x\) by performing operations equally on both sides. Importantly, when you multiply or divide by a negative number, reverse the inequality sign to maintain the correct relationship between the numbers involved.

After solving, you'll have conditions like \(x \leq 3\) from the first inequality and \(x \geq \frac{1}{3}\) from the second. The final solution combines these conditions by taking the intersection, resulting in \(\frac{1}{3} \leq x \leq 3\).

Expressing Solutions using Interval Notation

Once you have determined the solution to a compound inequality, expressing it in interval notation provides a clean, concise way to convey the range of values \(x\) can take. Interval notation captures the essence of the solution, highlighting the boundary points and whether they are included in the set.

For our inequality \(\frac{1}{3} \leq x \leq 3\), the solution set is written as \([\frac{1}{3}, 3]\). The square brackets \([\ ])\) indicate that \(\frac{1}{3}\) and \(3\) are included within the solution set, meaning the inequality is "inclusive." Inclusive inequalities are typical with compound inequalities that form a continuous span of numbers.

Interval notation is particularly useful because it's a standardized format that clearly communicates the solution set without needing lengthy explanations. It's easily readable and commonly used in mathematics to handle expressions that define variable ranges.

Visualizing with Graphing Inequalities

Graphing inequalities helps you visualize the solution set directly on a number line, showcasing the range of potential solutions. This visual representation is invaluable for comprehending which values satisfy the inequality and allows you to see the relationship between multiple inequalities at once.

To graph the solution \([\frac{1}{3}, 3]\), begin by drawing a number line with relevant points marked, such as \(\frac{1}{3}\) and \(3\). Since both endpoints of our interval are included in the solution, indicated by square brackets in interval notation, you place solid dots on these points.

Shade the region between these two points to visually highlight every number \(x\) that satisfies the inequality. This shaded area represents the continuous set of solutions. Graphs not only confirm the correctness of your algebraic solution but also provide an intuitive understanding of the variable's range. Seeing a problem solved both numerically and visually can enhance comprehension and retention.