Problem 24 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{2}{3}-\frac{1}{2} x \geq \frac{1}{6}+x $$

Step-by-Step Solution

Answer

Solution: \( (-\infty, \frac{1}{3}] \)

Step 1: Eliminate Fractions by Multiplying by the Least Common Denominator

The given inequality is \( \frac{2}{3} - \frac{1}{2}x \geq \frac{1}{6} + x \). To eliminate the fractions, let's find the Least Common Denominator (LCD) of the denominators 3, 2, and 6, which is 6. Multiply every term by 6:\[ 6 \times \left( \frac{2}{3} - \frac{1}{2}x \right) \geq 6 \times \left( \frac{1}{6} + x \right) \]This simplifies to:\[ 4 - 3x \geq 1 + 6x \]

Step 2: Move All Terms Involving x to One Side

Subtract \( 6x \) from both sides of the inequality:\[ 4 - 3x - 6x \geq 1 \]This gives:\[ 4 - 9x \geq 1 \]

Step 3: Isolate the x Term

Subtract 4 from both sides to move the constant term to the right side:\[ -9x \geq 1 - 4 \]This simplifies to:\[ -9x \geq -3 \]

Step 4: Solve for x by Dividing by the Coefficient of x

Divide each side by \(-9\) to solve for \(x\). Note that dividing by a negative number reverses the inequality:\[ x \leq \frac{-3}{-9} \]This simplifies to:\[ x \leq \frac{1}{3} \]

Step 5: Express the Solution in Interval Notation

The solution \( x \leq \frac{1}{3} \) in interval notation is written as:\[ (-\infty, \frac{1}{3}] \]

Step 6: Graph the Solution Set

To graph \( x \leq \frac{1}{3} \), draw a number line. Place a closed circle (or dot) on \( \frac{1}{3} \) and shade the region extending to the left to represent all solutions where \( x \leq \frac{1}{3} \).

Key Concepts

Least Common Denominator

When dealing with fractions in equations or inequalities, it can be quite troublesome to manage different denominators. A great way to simplify such problems is to find the Least Common Denominator (LCD). The LCD is the smallest number that each of the denominators can divide into evenly.

For example, consider the inequality \( \frac{2}{3} - \frac{1}{2}x \geq \frac{1}{6} + x \). Here, the denominators are 3, 2, and 6. The smallest number all three can divide into without leaving a remainder is 6, which is our LCD. Multiplying the entire equation by 6 eliminates the fractions easily:

• Multiply \( 6 \times \frac{2}{3} \) to get 4.
• Multiply \( 6 \times \frac{1}{2} \) and \( 6 \times x \), to remove the fractions next to \(x\).
• Finally, multiply \( 6 \times \frac{1}{6} \).

This manipulation serves to simplify the problem without changing the relationship between the variables, making it easier to solve.

Solution Set

The set of all possible values that satisfy the inequality is called the solution set. For the inequality \(4 - 9x \geq 1\), we initially work through individual steps to isolate \(x\). After simplifying and solving for \(x\), the inequality changes to \(x \leq \frac{1}{3}\).

This tells us that all values of \(x\) less than or equal to \(\frac{1}{3}\) form our solution set. A solution set might sometimes include an exact boundary point, such as \(\frac{1}{3}\) in this case, hence the inequality uses "less than or equal to".

By understanding the concept of a solution set, you can see which values satisfy your inequality, providing a clearer picture of its underlying meaning.

Interval Notation

Once you have identified the solution set, expressing it using interval notation is an organized method to communicate it effectively. In interval notation, a solution like \(x \leq \frac{1}{3}\) translates to \((-\infty, \frac{1}{3}]\).

Here's a breakdown:

• \((-\infty, \frac{1}{3}]\) indicates all numbers from negative infinity up to and including \(\frac{1}{3}\).
• The curved parenthesis "(" indicates that negative infinity is not a number that can be reached, just approached.
• The square bracket "]" implies that the number \(\frac{1}{3}\) is included in the solution set.

Using interval notation provides a concise way of listing all the potential solutions to an inequality, aiding in clearer mathematical communication.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. For the inequality \(x \leq \frac{1}{3}\), the graphing process involves a few simple steps.

The number line should extend from negative to positive values. Here's how you graph it:

• Locate the point \(\frac{1}{3}\) on the number line.
• Place a closed circle or dot at \(\frac{1}{3}\). A closed circle signals that \(\frac{1}{3}\) is included in the solution.
• Shade all points to the left of \(\frac{1}{3}\), heading towards negative infinity, indicating all these values are part of the solution set.

This visual representation helps you immediately understand where the solutions lie relative to \(\frac{1}{3}\), making it easier to interpret and communicate the inequality's solutions.