Problem 74 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(-3

Step-by-Step Solution

Answer

The solution set is \((-4, 11)\).

Step 1: Set Up the Compound Inequality

Start with the given compound inequality: \[-3 < f(x) < 7\]We know that \(f(x) = \frac{2}{3}x - \frac{1}{3}\). Substitute \(f(x)\) in the inequality:\[-3 < \frac{2}{3}x - \frac{1}{3} < 7\]

Step 2: Solve the Left Part of the Inequality

Handle the part \(-3 < \frac{2}{3}x - \frac{1}{3}\). Add \(\frac{1}{3}\) to both sides to begin isolating \(x\):\[-3 + \frac{1}{3} < \frac{2}{3}x\]Convert \(-3\) into a fraction: \[-\frac{9}{3} + \frac{1}{3} < \frac{2}{3}x\]Simplify:\[-\frac{8}{3} < \frac{2}{3}x\]

Step 3: Solve for x in the Left Inequality

To isolate \(x\), multiply both sides by \(\frac{3}{2}\):\[-\frac{4}{2} < x\]Simplify:\[-4 < x\]

Step 4: Solve the Right Part of the Inequality

Now handle the part \(\frac{2}{3}x - \frac{1}{3} < 7\). Add \(\frac{1}{3}\) to both sides:\[\frac{2}{3}x < 7 + \frac{1}{3}\]Convert \(7\) to a fraction:\[\frac{2}{3}x < \frac{21}{3} + \frac{1}{3}\]Simplify:\[\frac{2}{3}x < \frac{22}{3}\]

Step 5: Solve for x in the Right Inequality

Multiply both sides by \(\frac{3}{2}\) to solve for \(x\):\[x < \frac{22}{2}\]Simplify:\[x < 11\]

Step 6: Combine the Solutions

Combine the results from Steps 3 and 5:\[-4 < x < 11\]This is the solution to the compound inequality.

Step 7: Graph the Solution Set

Draw a number line. Plot open circles at \(-4\) and \(11\) to indicate that these endpoints are not included. Shade the region between \(-4\) and \(11\) to represent the set of all \(x\) values that satisfy the inequality.

Step 8: Write the Interval Notation

The solution represents the interval of \(x\) values from \(-4\) to \(11\), not including the endpoints. Write this in interval notation as:\((-4, 11)\)

Key Concepts

Interval Notation

When you solve an inequality, it's important to express the solution clearly. One fantastic way to do that is with interval notation. This type of notation is concise and makes it easy to visualize and understand the range of solutions.

Here's a quick rundown of interval notation:

• Use parentheses () for numbers that are not included in the solution set (this is also called an open interval).
• Use brackets [] for numbers that are included in the solution (a closed interval).
• The interval starts with the smallest number and goes to the largest.
• The union of two intervals can be expressed with a U symbol.

In our example, the solution was \(-4 < x < 11\). This is written in interval notation as \((-4, 11)\), meaning all numbers between -4 and 11 are solutions, but -4 and 11 themselves are not included.

Interval notation is a powerful and efficient way to represent inequality solutions, making complex problems look simple!

Solving Inequalities

Solving inequalities is a lot like solving equations, with a few key differences. The goal is to isolate the variable on one side of the inequality symbol. Let's break it down:

• Begin with the inequality, just like with an equation.
• To simplify, you can add, subtract, multiply, or divide both sides by the same number, just like in equations.
• Here's a critical part: If you multiply or divide by a negative number, flip the inequality sign!

In our exercise, we solved \(-3 < f(x) < 7\) by first substituting \(f(x) = \frac{2}{3}x - \frac{1}{3}\), then solving each part separately. The key steps involved isolating \(x\) and performing operations carefully to balance both sides of the inequality.

These steps lead us to the solution \(-4 < x < 11\), representing the range of numbers that satisfy the original compound inequality. Understanding how to apply these operations correctly is essential in the art of solving inequalities.

Graphing Inequalities

Graphing inequalities is a great way to visually understand where solutions lie. It involves plotting solutions on a number line. Let's dive into how this works:

• Start with a number line, a straight line where each point corresponds to a real number.
• Identify the critical points in your solution, such as the endpoints of intervals.
• Use open circles for numbers that are not included (like \(-4\) and \(11\) in our example) and closed circles for numbers that are included.
• Finally, shade the section of the number line between these points to represent all possible solution values.

For \(-4 < x < 11\), you would draw a number line with open circles at -4 and 11. Then shade the portion between them. This visualization helps you quickly see the range of numbers that are valid solutions.

Graphing is a powerful tool, especially useful in showing solution sets and indicating how they relate to real-world situations, making abstract math tangible and easier to grasp.