Problem 65 - Elementary and 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

The solution is \((-12, -6]\) and is represented with an interval on a number line.

Step 1: Isolate the Inequality Terms

Start with the compound inequality: \[-24 < \frac{3}{2}x - 6 \leq -15\]Add 6 to each part of the compound inequality to eliminate the constant term:\[-24 + 6 < \frac{3}{2}x \leq -15 + 6\]This simplifies to:\[-18 < \frac{3}{2}x \leq -9\]

Step 2: Solve for x

To isolate \(x\), divide every part of the inequality by \(\frac{3}{2}\) to solve for \(x\):\[\frac{-18}{\frac{3}{2}} < x \leq \frac{-9}{\frac{3}{2}}\]This further simplifies to:\[-12 < x \leq -6\]

Step 3: Express in Interval Notation

The solution to the inequality can be expressed in interval notation. Since \(x\) is greater than \(-12\) but less than or equal to \(-6\), the solution in interval notation is:\((-12, -6]\)

Step 4: Graph the Solution Set

To graph \((-12, -6]\):- Draw a number line.- Place an open circle at \(-12\) to represent that \(-12\) is not included in the solution.- Place a closed circle or a filled dot at \(-6\) to indicate that \(-6\) is included.- Shade the region between \(-12\) and \(-6\) to show all numbers in this range are included in the solution set.

Key Concepts

Solving inequalities

Solving inequalities is similar to solving equations, but there are special considerations. You're still solving for a variable, often represented as \(x\), by performing operations that maintain the inequality's balance. A compound inequality like \(-24 < \frac{3}{2}x - 6 \leq -15\) involves handling two inequality signs at once. This means you need to solve two inequalities at the same time to find the values of \(x\) that satisfy both conditions simultaneously.

Start by isolating the variable term. This often involves moving terms across the inequality, always ensuring consistency across all parts of the compound expression.

For instance, by adding \(6\) to each part of the original compound inequality, you get:

• \(-18 < \frac{3}{2}x \leq -9\)

This step simplifies the problem, making it easier to isolate \(x\).

Next, deal with coefficients by using multiplication or division. Here, dividing the entire inequality by \(\frac{3}{2}\), you simplify to:

• \(-12 < x \leq -6\)

Remember, dividing an inequality by a negative number would flip the inequality signs, but since \(\frac{3}{2}\) is positive, the signs remain unchanged. This crucial detail distinguishes solving inequalities from solving equations.

Interval notation

Understanding interval notation is key to expressing the solution to an inequality compactly. This form captures the range of values that solve the inequality without needing longer explanations or lists of numbers.

Consider an interval such as \((-12, -6]\). This tells you that \(x\) is greater than \(-12\) but less than or equal to \(-6\). It uses two types of brackets:

• Parentheses \(( )\) indicate the endpoint is not included.
• Square brackets \([ ]\) indicate the endpoint is included.

In this specific case, the parentheses at \(-12\) indicate \(-12\) itself is not a solution, while the square bracket at \(-6\) means \(-6\) is part of the solution set.

This simple format efficiently conveys the same information that could otherwise be cumbersome to write using words.

Graphing inequalities

Graphing inequalities provides a visual representation that can make solutions easier to understand at a glance. To graph the solution \((-12, -6]\), you use a number line. This visual tool helps map out the range of solutions effectively.

Start by determining the endpoints:

• \(-12\) should have an open circle because it's not included in the solution set.
• \(-6\) should have a closed circle to show it is included.

With those markers in place, shade the line between \(-12\) and \(-6\) to signify every number in this interval is a solution for the inequality.

This shaded region represents all possible values of \(x\) satisfying the compound inequality, allowing others to understand possible solutions quickly. Graphical solutions are advantageous in terms of conveying information swiftly and succinctly, and they can be particularly helpful when dealing with complex inequalities.