Problem 112 - A Graphical Approach to College Algebra

Question

Solve each three-part inequality analytically. Support your answer graphically. $$-2<\frac{x-4}{6}<2$$

Step-by-Step Solution

Answer

The solution is \(-8 < x < 16\).

Step 1: Break Down the Inequality

The given inequality is \(-2<\frac{x-4}{6}<2\). This is a compound inequality, which means it consists of two inequalities: \(-2 < \frac{x-4}{6}\) and \(\frac{x-4}{6} < 2\). We need to solve each part separately.

Step 2: Solve the First Inequality

First, solve \(-2 < \frac{x-4}{6}\):1. Multiply each side by 6 to eliminate the fraction: \(-2 \times 6 < x - 4\times 6\)2. Simplify to obtain \(-12 < x - 4\).3. Add 4 to both sides: \(-8 < x\).

Step 3: Solve the Second Inequality

Now, solve \(\frac{x-4}{6} < 2\):1. Multiply each side by 6: \(x - 4 < 2 \times 6\).2. Simplify to obtain \(x - 4 < 12\).3. Add 4 to both sides: \(x < 16\).

Step 4: Combine the Solutions

Combine the results from the two separate inequalities:- From Step 2, we have \(-8 < x\).- From Step 3, we have \(x < 16\).Thus, the combined inequality is \(-8 < x < 16\).

Step 5: Graph the Solution

On a number line, \(x\) falls between \(-8\) and \(16\), so draw an open circle at \(-8\) and \(16\), indicating that \(x\) does not include these boundary points. Shade the region between these two points to represent all possible solutions where \(-8 < x < 16\).

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities that are combined into a single statement. In this case, the inequality is

• \(-2<\frac{x-4}{6}<2\)

This means we need to consider not just one, but two expressions to solve for. They are linked by the shared variable, which in this case is \(x\).
Here's how it works:

• Solving one part of the inequality gives information about one side (e.g., \(-8 < x\)).
• Solving the other part gives information about the other side (e.g., \(x < 16\)).

The solutions of these inequalities are combined to form a range of values for \(x\). This range or interval indicates that \(x\) is greater than \(-8\) and less than \(16\). In simpler terms, \(x\) can be any number between \(-8\) and \(16\), excluding the endpoints.

Graphical Representation

Graphically representing compound inequalities helps in visualizing solutions.

• First, identify the range of \(x\) from the inequality. In our solved case, it is \(-8 < x < 16\).
• Draw a number line, marking crucial values such as \(-8\) and \(16\).

Open circles are used to indicate that the specific points are not included in the solution.
This is because the inequality signs are strict; they do not include \(-8\) and \(16\) in the range of \(x\).
Once properly marked, shade the region between the two points. This shaded portion represents all the possible values of \(x\) that satisfy both parts of the compound inequality.

Analytical Solution

The analytical approach involves systematically working through the algebraic expressions of the inequalities to find the solution. Here's how it typically unfolds:

• Start by isolating the variable in each part of the compound inequality.

For the inequality

• \(-2<\frac{x-4}{6}<2\)

first, solve the individual inequalities separately:

• Solve \(-2 < \frac{x-4}{6}\): clear the fraction by multiplying through by 6, leading to the solution \(-8 < x\).
• Solve \(\frac{x-4}{6} < 2\): similarly, multiply through by 6, leading to the solution \(x < 16\).

After finding solutions to both separate inequalities, combine them to form the full solution of the compound inequality:

• \(-8 < x < 16\)

This systematic approach ensures each part is treated and understood correctly, guaranteeing a thorough and precise solution.