Problem 19 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{1}{2} x-\frac{2}{3}>2 $$

Step-by-Step Solution

Answer

\( x > \frac{16}{3} \), interval: \( \left( \frac{16}{3}, \infty \right) \).

Step 1: Clear the Fractions

To eliminate the fractions, multiply every term in the inequality \( \frac{1}{2}x - \frac{2}{3} > 2 \) by the least common multiple (LCM) of the denominators, which in this case is 6. This yields: \( 6 \cdot \frac{1}{2}x - 6 \cdot \frac{2}{3} > 6 \cdot 2 \). Simplifying, we get \( 3x - 4 > 12 \).

Step 2: Isolate the Variable

Add 4 to both sides of the inequality \( 3x - 4 > 12 \) to move the constant term:\[ 3x - 4 + 4 > 12 + 4 \]This simplifies to:\[ 3x > 16 \]

Step 3: Solve for the Variable

Divide both sides of the inequality \( 3x > 16 \) by 3 to solve for \( x \):\[ x > \frac{16}{3} \]This represents the solution in terms of \( x \).

Step 4: Represent in Interval Notation

The solution \( x > \frac{16}{3} \) in interval notation is \( \left( \frac{16}{3}, \infty \right) \). This indicates that \( x \) can be any number greater than \( \frac{16}{3} \).

Step 5: Graph the Solution Set

On a number line, draw an open circle at \( \frac{16}{3} \) to show that this point is not included in the solution. Then, draw a line extending to the right to represent all numbers greater than \( \frac{16}{3} \).

Key Concepts

Interval Notation

When solving inequalities, expressing the solution using interval notation provides a concise way to identify the set of all possible solutions. Interval notation uses parentheses and brackets to show which numbers are included or excluded in the solutions.
For example, if you have a solution like \( x > \frac{16}{3} \), in interval notation it is expressed as \( \left( \frac{16}{3}, \infty \right) \). The parenthesis \(( )\) always indicates that the endpoint is not included in the set, while a bracket \([ ]\) would mean the endpoint is included.

• \((a, b)\) means greater than \(a\) and less than \(b\), excluding both endpoints.
• \((a, \infty)\) means greater than \(a\), extending to infinity, excluding \(a\).
• \([-\infty, a)\) means less than \(a\), extending to negative infinity, excluding \(a\).

This notation is very useful for clearly communicating solutions of inequalities in an easy-to-read format.

Graphing Inequalities

Graphing inequalities on a number line is a straightforward way to visualize the solution set. In our example of solving the inequality \( x > \frac{16}{3} \), graphing shows which numbers satisfy the inequality.
Here's how you would graph this inequality:

• First, draw a number line with appropriate markings around the critical value, in this case, \( \frac{16}{3} \).
• Next, place an open circle on \( \frac{16}{3} \) to show that this number is not included in the solution set.
• Then, draw a line extending to the right from \( \frac{16}{3} \) to show that all numbers greater than \( \frac{16}{3} \) meet the condition \( x > \frac{16}{3} \).

The open circle indicates that \( x \) values can get infinitely close to \( \frac{16}{3} \) but never actually reach it. Thus, graphically representing inequalities helps in understanding which areas on the number line are part of the solution.

Solving Inequalities

Solving linear inequalities involves a few systematic steps similar to solving equations. The key is to isolate the variable while keeping the inequality balanced. Let's understand the approach with our example inequality \( \frac{1}{2}x - \frac{2}{3} > 2 \).

• Clear Fractions: Begin by eliminating fractions for simpler operations. Here, multiply each term by the LCM of denominators, which is 6, giving \( 3x - 4 > 12 \).
• Isolate the Variable: Add or subtract terms to move constants to one side. For \( 3x - 4 > 12 \), adding 4 to both sides results in \( 3x > 16 \).
• Solve for the Variable: Divide by the coefficient of \( x \) to solve. Dividing both sides by 3 gives \( x > \frac{16}{3} \).

One important precaution: when multiplying or dividing an inequality by a negative number, the inequality sign must be flipped. However, this is not necessary in this case as we dealt with positive numbers. Practice and repetition make solving inequalities second nature.