Problem 84 - Intermediate Algebra

Question

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. \(\frac{x-3}{2} \leq \frac{1}{2}-\frac{x-5}{4}\) b. \(\frac{x-3}{2}>\frac{1}{2}-\frac{x-5}{4}\)

Step-by-Step Solution

Answer

a. \((-\infty, \frac{13}{3}]\); b. \((\frac{13}{3}, \infty)\).

Step 1: Clear Fractions

To solve the inequality \(\frac{x-3}{2} \leq \frac{1}{2} - \frac{x-5}{4}\), start by eliminating fractions by finding the least common denominator (LCD) which is 4. Multiply each term by 4 to get:\[4 \cdot \frac{x-3}{2} \leq 4 \cdot \left(\frac{1}{2} \right) - 4 \cdot \frac{x-5}{4}\] which simplifies to: \[2(x-3) \leq 2 - (x-5)\].

Step 2: Distribute and Simplify

Distribute the numbers in the expression: \[2(x-3) = 2x - 6\] and for the other side: \[-(x-5) = -x + 5\]. This gives us the simplified inequality:\[2x - 6 \leq 2 - x + 5\].

Step 3: Combine Like Terms

Rewrite the inequality by combining like terms:\[2x - 6 \leq -x + 7\]. Then, add \(x\) to both sides to get:\[3x - 6 \leq 7\].

Step 4: Solve for x

Add 6 to both sides to isolate terms with \(x\):\[3x \leq 13\].Divide by 3 to solve for \(x\):\[x \leq \frac{13}{3}\].

Step 5: Write Solution in Interval Notation

The solution to the inequality is \(x \leq \frac{13}{3}\). In interval notation, this is written as:\((-\infty, \frac{13}{3}]\).

Step 6: Graph Solution Set

To graph the solution \(x \leq \frac{13}{3}\), draw a number line, mark the point \(\frac{13}{3}\) with a filled circle, and shade the region to the left, indicating all values less than or equal to \(\frac{13}{3}\).

Step 7: Determine Solution for Part b

The solution to part b's inequality \(\frac{x-3}{2} > \frac{1}{2} - \frac{x-5}{4}\) is the opposite of part a, because it's a strict inequality. Therefore, it is \(x > \frac{13}{3}\).

Step 8: Write Solution for Part b in Interval Notation

The solution for part b is \(x > \frac{13}{3}\), which in interval notation is:\( \left(\frac{13}{3}, \infty \right)\).

Step 9: Graph Solution Set for Part b

To graph the solution \(x > \frac{13}{3}\), draw a number line, mark the \(\frac{13}{3}\) point with an open circle, and shade the area to the right, indicating all values greater than \(\frac{13}{3}\).

Key Concepts

Solving Inequalities

Whenever you're dealing with inequalities, the goal is to find which range of values for the variable makes the inequality true. Inequalities are a bit like equations, but with inequalities, we use symbols like \(<, >, \leq, \geq\) instead of the \(=\) sign. Here's how you solve them in a systematic way:

• Start by removing any fractions if they exist. This helps to simplify the inequality to more manageable expressions. You can do this by multiplying every term by the least common denominator (LCD).
• Next, distribute any numbers across terms within brackets and simplify the expression by combining like terms. This makes sure that all the variable terms and constants are in their simplest form.
• Then, isolate the variable on one side of the inequality by adding or subtracting terms from both sides. Keep in mind, if you multiply or divide by a negative number, you have to flip the inequality sign!
• The last step is solving the simplified inequality for the variable, which typically leaves you with a solution that looks like \(x \leq \frac{13}{3}\).

This process transforms a complex looking inequality into a simple statement about a range of numbers.

Interval Notation

Once you've found the solution to an inequality, it's important to express it in a clean and concise way. This involves using something called interval notation. It's a mathematical shorthand that uses parentheses and brackets to describe ranges of numbers:

• If the range of values includes the endpoint, use a closed bracket \([ ]\). This happens with inequalities that use \(\leq\) or \(\geq\).
• If the endpoint isn't included, you use a parenthesis \(( )\). This is used for strict inequalities symbolized by \(<\) or \(>\).
• For values extending to infinity, always use a parenthesis, because infinity isn't a number you can reach and "close" off.

For instance, if \(x \leq \frac{13}{3}\), the interval notation is \(( -\infty, \frac{13}{3}]\). If it was \(x > \frac{13}{3}\), you'd write that as \( \left( \frac{13}{3}, \infty \right) \). This format provides a quick comprehension of the solution set for any inequality.

Graphing Solutions

Visualizing the solution of an inequality can really make understanding easier. Graphing is all about showing which parts of the number line are included in the solution:

• Start by drawing a simple number line, marking key points, especially where the boundary values of the inequality lie.
• Use a filled circle at the boundary if the value is included (\(\leq\) or \(\geq\)). An open circle is used if it's not included (\(<\) or \(>\)).
• Shade the region of the number line that corresponds to the solution set. This helps in visualizing not just the boundary, but also the range of solutions.

For example, with the inequality \(x \leq \frac{13}{3}\), place a filled circle on \(\frac{13}{3}\) and shade everything to the left. While for \(x > \frac{13}{3}\), use an open circle on \(\frac{13}{3}\) with shading to the right. This represents how the solutions spill across the number line.