Problem 105 - A Graphical Approach to College Algebra

Question

Solve each three-part inequality analytically. Support your answer graphically. $$-10>3 x+2>-16$$

Step-by-Step Solution

Answer

\(-6 < x < -4\).

Step 1: Break Down the Compound Inequality

The given inequality is \(-10 > 3x + 2 > -16\). This is a compound inequality, which means it consists of two inequalities combined. The task is to solve each inequality separately, i.e., \(-10 > 3x + 2\) and \(3x + 2 > -16\).

Step 2: Solve the First Inequality

For the first part, \(-10 > 3x + 2\), subtract 2 from both sides to isolate the term with the variable:\[-10 - 2 > 3x\]\[-12 > 3x\].Next, divide by 3 to solve for \(x\):\[-4 > x\].

Step 3: Solve the Second Inequality

For the second part, \(3x + 2 > -16\), subtract 2 from both sides to isolate the term with the variable:\[3x > -16 - 2\]\[3x > -18\].Next, divide by 3 to solve for \(x\):\[x > -6\].

Step 4: Combine the Solutions

Now, combine the results of both inequalities. From Step 2, we have \(-4 > x\), and from Step 3, \(x > -6\). This means:\[-6 < x < -4\].

Step 5: Graph the Solution on a Number Line

Draw a number line and plot the interval \(-6 < x < -4\). This is an open interval, so use open circles at -6 and -4 to indicate that these numbers are not included in the solution. Shade the region between these points to represent all possible solutions for \(x\).

Key Concepts

Understanding Compound Inequalities

Compound inequalities are mathematical expressions where two or more individual inequalities are combined into a single statement. These are typically connected by the words "and" or "or." In our exercise, the inequality \(-10 > 3x + 2 > -16\) is an example of a compound inequality. It implies two separate inequalities combined into one:

• First inequality: \(-10 > 3x + 2\)
• Second inequality: \(3x + 2 > -16\)

The goal is to find the values of \(x\) that satisfy both conditions at the same time. Essentially, we're seeking an overlap in the solutions of the two separate inequalities.
To manage compound inequalities, it's often a good practice to break them down into the individual inequalities as seen in our exercise solution. Decomposing them allows for straightforward solving, step by step, for each component part.

Graphical Representation of Solutions

Graphing solutions to inequalities is important because it allows you to visualize the range of solutions easily. For a compound inequality like \(-6 < x < -4\), the best way to represent the solution is on a number line. Here's how you can graph this specific inequality:

• Draw a horizontal line to represent the number line.
• Identify the numbers involved in your solution: -6 and -4.
• Place open circles at -6 and -4 to show that these endpoints are not included in the solution (since our solution doesn’t include them).
• Shade the region between -6 and -4. This shaded area represents all values of \(x\) that satisfy both inequalities.

The open circles indicate that these boundary numbers are not included. The shaded region between them illustrates the solutions that work for both inequalities at once, enhancing understanding through visual means.

Solving Inequalities with the Analytical Method

An analytical approach involves solving equations step by step using algebraic manipulations to find the solution set. It is an essential skill for understanding the behavior of inequalities.
In the given example, we start by isolating the variable \(x\) in each part of the compound inequality:

• Isolate \(x\) for the first inequality \(-10 > 3x + 2\): subtract 2 from each side and then divide by 3, resulting in \(x < -4\).
• Repeat the process for the second inequality \(3x + 2 > -16\): subtract 2 from each side and divide by 3, giving \(x > -6\).
• Combine these results to form a single range: \(-6 < x < -4\), which represents all possible solutions for the original compound inequality.

This method relies on a systematic approach to isolate the variable and manipulate the inequality using basic algebraic rules. It's a clean and logical way to determine all possible solutions for a compound inequality, verifying the solution with further visual tools like graphs for enhanced clarity.