Problem 71 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(x \leq 6-\frac{1}{2} x\) and \(\frac{1}{2} x+1 \geq 3\)

Step-by-Step Solution

Answer

The solution is \( x = 4 \); in interval notation: \( \{4\} \).

Step 1: Solve the first inequality

The first inequality given is \( x \leq 6 - \frac{1}{2}x \). First, combine like terms by adding \( \frac{1}{2}x \) to both sides: \( x + \frac{1}{2}x \leq 6 \). This simplifies to \( \frac{3}{2}x \leq 6 \). Divide both sides by \( \frac{3}{2} \) to isolate \( x \): \( x \leq 4 \).

Step 2: Solve the second inequality

For the second inequality \( \frac{1}{2}x + 1 \geq 3 \), start by subtracting 1 from both sides to get \( \frac{1}{2}x \geq 2 \). Multiply both sides by 2 to solve for \( x \): \( x \geq 4 \).

Step 3: Determine the intersection of solutions

The solutions to the inequalities are \( x \leq 4 \) and \( x \geq 4 \). The intersection of these solutions is \( x = 4 \).

Step 4: Graph the solution

The solution \( x = 4 \) is a single point on the number line. You can graph this by placing a solid dot at 4 to indicate that this value is included in the solution set.

Step 5: Write the solution in interval notation

The solution \( x = 4 \) in interval notation is \( \{4\} \), indicating the set containing only the number 4.

Key Concepts

Solution Set

In mathematics, a solution set is the collection of all possible values that satisfy a given equation or inequality. When working with compound inequalities, we typically deal with two or more inequalities combined together. For this specific problem, we had two inequalities:

• \(x \leq 4\)
• \(x \geq 4\)

The solution to these inequalities is the overlap or intersection of all possible solutions, where both conditions hold true at the same time. For our example, the solution set is quite specific: the number 4. A solution set indicates exactly which values are part of the solution, allowing us to express these values concisely. For a single specific solution like this, we could express the solution set in set notation as \(\{4\}\), showing that 4 is the number that fulfills all given inequalities simultaneously. This approach helps us clearly understand the boundary or the exact values where the inequalities intersect.
Understanding solution sets is essential in problem-solving since it defines the exact values satisfying the given conditions in mathematical problems.

Graphing Inequalities

Graphing inequalities allows us to visually represent the solution set of an inequality or a compound inequality on a number line or coordinate plane. When graphing, we use open dots to indicate values that are not part of the solution and solid dots for values that are included.
In our exercise, we found that the solution is \(x = 4\). To graph this on a number line:

• Place a solid dot on 4 to indicate that this value is included in the solution set.

This single dot shows the condition "\(x \leq 4\) and \(x \geq 4\)", meaning that out of both inequalities only 4 is the solution. This is straightforward as the solution intersects at a single point – one solid dot.
By using graphs, we simplify complex algebraic expressions into easy-to-understand visual elements that clearly depict the range or exact points of solutions.

Interval Notation

Interval notation offers a concise way of representing a set of numbers that are solutions to an inequality or a group of inequalities. It uses brackets and parentheses to show whether endpoints are included or not.
In our specific solution, the only number in the solution set is 4. Thus, the interval notation becomes \(\{4\}\), which is technically set notation as it's better suited for single values rather than ranges.
Interval notation typically looks like this:

• \([a, b]\): Includes both endpoints a and b (inclusive).
• \((a, b)\): Excludes both endpoints a and b (exclusive).
• \([a, b)\) or \((a, b]\): Includes one endpoint but not the other.

However, in situations like ours where the solution is a single point rather than a range, we represent it within braces \(\{ \, \} \) to indicate it's just that single element. Understanding interval and set notation helps in communicating solutions effectively in algebra and calculus.