Problem 22 - Algebra and Trigonometry

Question

\(9-32\) me 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

The solution is \((-\infty, \frac{1}{3}]\).

Step 1: Eliminate Fractions

Multiply every term by the Least Common Denominator (LCD) of the fractions, which is 6, to eliminate the fractions from the inequality: \[ 6 \left( \frac{2}{3} \right) - 6 \left( \frac{1}{2}x \right) \geq 6 \left( \frac{1}{6} \right) + 6x \] \Simplifying each term, we get: \[ 4 - 3x \geq 1 + 6x \]

Step 2: Move All Terms to One Side

Rearrange the inequality by getting all terms involving \(x\) to one side and constant terms to the other side: \[ 4 - 1 \geq 6x + 3x \] This simplifies to: \[ 3 \geq 9x \]

Step 3: Solve for x

Divide each side by 9 to solve for \(x\): \[ \frac{3}{9} \geq x \] Which simplifies to: \[ \frac{1}{3} \geq x \] Or equivalently: \[ x \leq \frac{1}{3} \]

Step 4: Express Solution in Interval Notation

Express the solution \(x \leq \frac{1}{3}\) in interval notation, which is: \[ (-\infty, \frac{1}{3}] \]

Step 5: Graph the Solution Set

On a number line, graph the solution \((-\infty, \frac{1}{3}]\): 1. Draw a line with an open circle starting at \(-\infty\) and a closed circle at \(\frac{1}{3}\). 2. Shade the line to the left of the closed circle to indicate all numbers less than or equal to \(\frac{1}{3}\) are included.

Key Concepts

Interval Notation

Interval notation is an efficient way of describing a set of numbers that fall within a particular range. In the context of inequalities, it conveys solutions succinctly.
In interval notation, brackets and parentheses are used to indicate whether endpoints are included or not. For example:

• A bracket, \([a, b]\), means both endpoints \(a\) and \(b\) are included in the set.
• A parenthesis, \(a, b)\), implies that the set includes numbers between \(a\) and \(b\), but not \(a\) or \(b\) themselves.
• Infinity, \(∞\), whether positive or negative, is represented with a parenthesis, like \((-∞, b]\) or \(a, ∞)\), because infinity is a concept, not a number.

In the exercise, the solution obtained was \(x \leq \frac{1}{3}\). This translates into interval notation as \((-∞, \frac{1}{3}]\), indicating every number from negative infinity up to and including \(\frac{1}{3}\).

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set's range, making it easier to comprehend which numbers satisfy the inequality. Here's how you can graphically represent, for example, the inequality \(x \leq \frac{1}{3}\):

• First, draw a horizontal line representing all possible values of \(x\).
• Locate the point \(\frac{1}{3}\) on the number line.
• Use a closed circle or dot at \(\frac{1}{3}\) since \(\frac{1}{3}\) is included in the solution set, as indicated by the "equal" part of the inequality.
• Shade the line to the left of \(\frac{1}{3}\) towards negative infinity, signaling all numbers less than or equal to \(\frac{1}{3}\).

Graphing inequalities is particularly important, as it gives a visual context that complements the symbolic solution, effectively conveying how expansive the solution set is.

Solving Inequalities Step by Step

When solving linear inequalities, follow methodical steps similar to solving equations. However, be cautious with inequality signs, especially when multiplying or dividing by negative numbers. Here's a structured approach using the example given:- **Step 1: Eliminate Fractions**: Multiply all terms by the Least Common Denominator (LCD) to clear fractions, simplifying calculations. In the example, multiplying by 6 helped transform the equation to \(4 - 3x \geq 1 + 6x\).
- **Step 2: Rearrange Terms**: Gather variable terms on one side and constants on the other. This reveals \(3 \geq 9x\), setting the stage for isolating the variable.
- **Step 3: Solve for the Variable**: Isolate \(x\) by dividing by the coefficient \(9\), ensuring no change to the inequality sign as the division is by a positive number, leading to \(x \leq \frac{1}{3}\).
While solving, remember:

• When multiplying/dividing by a negative number, flip the inequality sign.
• Double-check each step to ensure correct simplification and manipulation.

By following these steps, students can reliably solve similar inequalities, building both skill and confidence.