Problem 52 - 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. $$ x-2 \leq \frac{1}{3} x $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 3]\).

Step 1: Move All Terms to One Side

The first step is to move all terms to one side of the inequality. Start by subtracting \( \frac{1}{3}x \) from both sides to get one side equal to zero:\[x - 2 - \frac{1}{3}x \leq 0\]

Step 2: Simplify the Inequality

Combine the \( x \) terms by writing \(x\) as \(\frac{3}{3}x\):\[\frac{3}{3}x - \frac{1}{3}x - 2 \leq 0\]This simplifies to:\[\frac{2}{3}x - 2 \leq 0\]

Step 3: Solve for x

Add 2 to both sides to isolate the term with \( x \):\[\frac{2}{3}x \leq 2\]Next, multiply both sides by \(\frac{3}{2}\) to solve for \(x\):\[x \leq 3\]

Step 4: Write the Solution in Interval Notation

The solution to the inequality \(x \leq 3\) can be expressed in interval notation as:\[(-\infty, 3]\]

Step 5: Verify the Solution Symbolically

Verify the solution by substituting any number less than or equal to \(3\) in the inequality and ensure it holds true. For example, if \(x = 3\), the inequality becomes:\[3 - 2 \leq \frac{1}{3} \times 3 \]\[1 \leq 1\]Since the inequality holds true, the solution is verified.

Key Concepts

X-Intercept Method

The x-intercept method is a strategy used to solve inequalities by finding the value of \( x \) that makes the expression equal to zero. This method sets up an equation by moving all terms to one side, looking much like finding the root of an equation. Here’s how it works:

• Step 1: Rearrange the Inequality
  In our problem, we start with \( x - 2 \leq \frac{1}{3}x \). To use the x-intercept method, get all terms involving \( x \) on one side of the equation. Subtract \( \frac{1}{3}x \) from both sides to have: \( x - 2 - \frac{1}{3}x \leq 0 \).
• Step 2: Simplify the Expression
  Combine like terms, turning \( x - \frac{1}{3}x \) into \( \frac{2}{3}x \). The inequality now reads \( \frac{2}{3}x - 2 \leq 0 \).
• Step 3: Solve for \( x \)
  Add 2 to both sides to isolate \( \frac{2}{3}x \), giving \( \frac{2}{3}x \leq 2 \). Then, multiply both sides by the reciprocal, \( \frac{3}{2} \), to solve for \( x \): \( x \leq 3 \).

This result, \( x \leq 3 \), means that the inequality holds for every \( x \) value less than or equal to 3, which is the solution we find using the x-intercept method.

Interval Notation

Interval notation is a shorthand used in mathematics to express the set of solutions to inequalities. It provides a compact way to present a range of numbers that satisfy the inequality. Let’s explore how it applies to the solution of the problem \( x \leq 3 \):

• Understanding the Notation
  An interval is denoted by parentheses and/or brackets. Parentheses \(( )\) indicate that the endpoint is not included, while brackets \([ ]\) show that the endpoint is included in the set.
• Applying to the Solution
  For our inequality \( x \leq 3 \), \( 3 \) is included in the solution, thus shown with a bracket. The other side extends infinitely in the negative direction, represented by \(-\infty\), which is never included, so it uses a parenthesis. The interval notation for the solution is: \(( -\infty, 3 ]\).

Using interval notation makes it easy to quickly grasp which values \( x \) can take, and it’s particularly useful when graphically representing solutions on a number line.

Symbolic Verification

Symbolic verification is an essential step to confirm that a found solution satisfies the original inequality. This involves substituting test values from the solution set back into the initial inequality to double-check its validity.

• Choose a Test Value
  From our solution \( x \leq 3 \), we need to pick any value of \( x \) within this range. A simple choice is \( x = 3 \), the upper boundary of our interval.
• Substitute and Validate
  Insert this value into the original inequality: \( 3 - 2 \leq \frac{1}{3} \times 3 \). Simplifying this results in \( 1 \leq 1 \), a true statement.
• Confirm with Additional Values
  To be thorough, you might check other values, such as \( x = 0 \). Doing so, \( 0 - 2 \leq \frac{1}{3} \times 0 \) simplifies to \(-2 \leq 0 \), also true.

By confirming the inequality holds for test values, you add an essential layer of accuracy to ensure your solution is comprehensive and correct. Symbolic verification is a powerful tool to dismiss doubt and affirm confidence in your results.