Problem 51 - College Algebra with Modeling and Visualization

Question

Use the \(x\) -intercept method to solve the inequality. Write the solution set in set-builder or interval notation. Then solve the inequality symbolically. $$ x-3 \leq \frac{1}{2} x-2 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, 2]\).

Step 1: Rewrite the Inequality

The initial inequality is \( x - 3 \leq \frac{1}{2}x - 2 \). First, we need to get all \(x\) terms on one side and constants on the other. Subtract \(\frac{1}{2}x\) from both sides:\[ x - \frac{1}{2}x - 3 \leq -2 \]

Step 2: Simplify the Expression

Simplify \(x - \frac{1}{2}x\) on the left side. This becomes:\[ \frac{1}{2}x - 3 \leq -2 \]

Step 3: Isolate the Variable Term

Add 3 to both sides of the inequality to move the constant term to the right:\[ \frac{1}{2}x \leq 1 \]

Step 4: Solve for x

Multiply both sides of the inequality by 2 to solve for \(x\):\[ x \leq 2 \]

Step 5: Express the Solution in Interval Notation

The solution \(x \leq 2\) is represented in interval notation as:\( (-\infty, 2] \)

Key Concepts

x-intercept method

When solving inequalities like the one given, we use the x-intercept method as a visual technique because it can sometimes make inequalities more intuitive to understand. The key idea is to consider where the expression equals zero (the intercept) and check different regions around this point. However, keep in mind that not every inequality will have a direct x-intercept, as this method is more traditionally used for quadratic equations where you find the roots or zeroes. But we can still apply the concept slightly when solving linear inequalities:

• Consider each side of the inequality as a separate function.
• Set the expressions equal to each other to mimic the way you would find an x-intercept.
• Solve the equation symbolically as demonstrated, to find the critical point (here it's where the inequality turns into an equation).

In this exercise, the symbolic approach naturally brings us to the solution but involves conceptualizing it as points along the number line to decide where the inequality "switches".

interval notation

Interval notation is a concise way of describing a set of numbers along a number line. It uses parentheses and brackets to indicate which numbers are included in or excluded from the set. Understanding how to express solutions in this format is crucial for clearly communicating which values satisfy an inequality:

• Parentheses "))",