91Ó°ÊÓ

Solving inequalities is similar to solving equations, but with a key difference: inequalities show a range of solutions rather than a single solution. In this exercise, we start with the inequality:

\(3x+4 > \frac{1}{3}(x-2)\)

The goal is to find all values of \(x\) that satisfy this inequality.

Step-by-step Approach:

• Distribute the Fraction: Convert \(\frac{1}{3}(x-2)\) to \(\frac{1}{3}x - \frac{2}{3}\).
• Eliminate the Fraction: Multiply every term by 3 to rid of the fraction.
• Combine Like Terms: Move all \(x\)-terms to one side.
• Isolate the Variable:\(x\): Simplify the inequality to get \(x\) on one side.
• Solve for \(x\): Divide both sides by 8 to solve for \(x\).

By following these steps correctly, you find that the solution is \(x > -\frac{7}{4}\).

Once you've solved an inequality, you need to express the solution clearly. Interval notation is a method that succinctly shows the range of solutions.

For the inequality \(x > -\frac{7}{4}\), the solution is all numbers greater than \(-\frac{7}{4}\). We write this as:

\((-\frac{7}{4}, \infty)\)

Understanding Interval Notation:

• Parentheses \(()\): Indicate that the endpoint is not included in the interval. In our case, \(-\frac{7}{4}\) is not included because it's a strict inequality (\()>\).
• Infinity \(\infty\): Represents that there's no upper limit to our solution. Always use parentheses, never brackets, with infinity.

Interval notation provides a clear, concise way to communicate the range of solutions to an inequality.

Graphing inequalities involves shading a portion of the number line to represent the range of solutions. For the solution \(x > -\frac{7}{4}\), here's how to graph it:

• Draw the Number Line: Extend the line in both directions.
• Mark the Endpoint: Locate \(-\frac{7}{4}\) (~-1.75) and draw an open circle at this point. The open circle indicates that \(-\frac{7}{4}\) is not included in the solution.
• Shade the Interval: Shade the portion of the number line to the right of the open circle. This shows all values greater than \(-\frac{7}{4}\).

Graphing is a visual method to represent solutions and helps to understand the range of numbers that satisfy the inequality.