Problem 65 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(-24<\frac{3}{2} x-6 \leq-15\)

Step-by-Step Solution

Answer

Solution in interval notation: \((-12, -6]\)

Step 1: Understand the Compound Inequality

The given inequality is a compound inequality, which means we have two inequalities combined. It can be written as \(-24 < \frac{3}{2}x - 6\) and \(\frac{3}{2}x - 6 \leq -15\). We need to solve these two inequalities separately and then find the intersection of their solution sets.

Step 2: Solve the Left Inequality

Start with the inequality \(-24 < \frac{3}{2}x - 6\).First, add 6 to both sides to isolate the term with \(x\):\[-24 + 6 < \frac{3}{2}x \Rightarrow -18 < \frac{3}{2}x\]Next, multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\), to solve for \(x\):\[-18 \times \frac{2}{3} < x \Rightarrow x > -12\].

Step 3: Solve the Right Inequality

Now solve \(\frac{3}{2}x - 6 \leq -15\).First, add 6 to both sides:\[\frac{3}{2}x - 6 + 6 \leq -15 + 6 \Rightarrow \frac{3}{2}x \leq -9\]Next, multiply both sides by \(\frac{2}{3}\) to solve for \(x\):\[x \leq -9 \times \frac{2}{3} \Rightarrow x \leq -6\].

Step 4: Combine the Solutions

Combine the solutions from Step 2 and Step 3. We have:\(x > -12\) from Step 2 and \(x \leq -6\) from Step 3. To find the compound solution, take the intersection of these two:\(-12 < x \leq -6\).

Step 5: Write the Solution in Interval Notation

Express the solution \(-12 < x \leq -6\) in interval notation. This corresponds to:\((-12, -6]\).

Step 6: Graph the Solution

The graph of the solution \((-12, -6]\) is a number line:- Draw an open circle at \(-12\), indicating \(x\) cannot equal \(-12\).- Draw a closed circle at \(-6\), showing \(x\) can equal \(-6\).- Shade the region between these two points to represent all possible values of \(x\).

Key Concepts

Interval Notation

Interval notation is a way of writing the solution set of inequalities. It helps to succinctly express a range of values that a variable can take. When solving compound inequalities, interval notation allows you to represent the solution in a compact form.

In interval notation, brackets—or parentheses—are used to describe the set:

• Round brackets, \(( \text{ and } )\), indicate that the number next to it is not included in the set. This is often used with open inequalities like \(x > -12\).
• Square brackets, \([ \text{ and } ]\), show that the number is included. This is used with closed inequalities like \(x \leq -6\).

In our exercise, the solution from the compound inequalities is \(-12 < x \leq -6\). This translates to an interval notation of \((-12, -6]\), signifying that while \(x\) is greater than \(-12\), it can include \(-6\) as it is less than or equal to it. Understanding how to translate inequalities into interval notation is crucial for expressing solutions clearly.

Solving Inequalities

Solving inequalities involves finding the set of values that satisfy the given inequalities. It resembles solving equations but with additional rules for applications such as multiplying or dividing by negative numbers.

For our compound inequality \(-24 < \frac{3}{2}x - 6 \leq -15\), we isolated terms involving \(x\) in two separate parts. The process includes:

• Adding or subtracting terms to both sides to simplify the inequality expression.
• Multiplying or dividing the entire inequality by a positive number, maintaining the inequality's direction. If it's by a negative number, reverse the inequality sign.

By solving the separate inequalities, you can find where their solutions overlap, revealing the range of solutions for the compound inequality. Remember that each step must keep the inequality balance just like with equations, ensuring accurate outcomes.

Number Line Graphing

Graphing solutions on a number line provides a visual representation of an inequality solution, making it easier to understand how solutions are spread across numbers.

Once you determine the solution to a compound inequality, like \((-12, -6]\), graphing involves:

• Drawing a line to represent all real numbers.
• Using open circles to indicate numbers that are not part of the solution set (\(x\) does not include \(-12\)).
• Using closed circles to show numbers that are in the solution set (\(x\) does include \(-6\)).
• Shading the segment between these points to illustrate the complete solution range.

This visualization helps in quickly verifying or communicating the solution, showing not only the values that satisfy the inequality but also those that are left out. This method is particularly useful for educators and students to understand the extent of solution sets for inequalities.