Problem 72 - Algebra A Combined Function

Question

Solve each inequality. See Section 2.7. $$ \frac{z-2}{12}<\frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \( z < 5 \).

Step 1: Understand the Inequality

The given inequality is \( \frac{z-2}{12}<\frac{1}{4} \). We aim to find the values of \( z \) that satisfy this inequality.

Step 2: Eliminate the Denominator

Multiply both sides of the inequality by 12 to eliminate the denominator. This gives us:\[ z - 2 < \frac{12}{4} \]

Step 3: Simplify the Right Side

Simplify the fraction on the right side:\[ \frac{12}{4} = 3 \]So the inequality becomes:\[ z - 2 < 3 \]

Step 4: Solve for z

Add 2 to both sides of the inequality to solve for \( z \):\[ z < 3 + 2 \]

Step 5: Simplify the Expression

Complete the addition on the right side:\[ z < 5 \]

Step 6: Solution Check

Verify the solution by choosing a number less than 5, such as 4. Substitute back into the original inequality:\[ \frac{4 - 2}{12} < \frac{1}{4} \]\[ \frac{2}{12} = \frac{1}{6} \]Since \( \frac{1}{6} < \frac{1}{4} \), the solution is correct.

Key Concepts

Solving Inequalities

When working with inequalities, the goal is to determine the range of values that satisfy a particular inequality statement. In this context, inequalities compare two expressions using one of the following signs: \(<\), \(>\), \(\leq\), or \(\geq\). To solve an inequality, similar principles as solving equations are applied, with an extra precaution regarding multiplication or division by negative numbers, which reverses the inequality sign.

Let's consider the provided inequality \(\frac{z-2}{12}<\frac{1}{4}\). Solving inequalities often involves isolating the variable on one side, as shown in the example steps, starting with eliminating fractions by multiplying through by the denominator. This step transforms the problem into a simpler format, making it easier to proceed.

In this specific problem, the process is straightforward: multiply both sides by 12 to remove the denominator, simplifying expressions, and then solve for \(z\) using basic arithmetic operations like addition or subtraction.

Rational Inequalities

Rational inequalities involve ratios of algebraic expressions, such as \(\frac{z-2}{12}<\frac{1}{4}\). These types of inequalities typically require special attention because of the presence of fractions.

A rational inequality is solved by applying similar techniques used in solving regular inequalities: simplifying the expressions, and in some cases, finding a common denominator or clearing fractions to make the inequality manageable. It's important to be cautious about the inequality sign when multiplying or dividing by negative numbers, as it will flip the inequality direction, although in this example, this step wasn't needed.

This exercise focused on clearing the fractional expression by directly multiplying by the denominator, which is effective when the denominator is constant and non-zero in value. This transformation often simplifies the inequality significantly, allowing you to focus on solving the linear form.

Inequalities Solution Verification

Once an inequality solution is derived, it's vital to verify that the solution actually satisfies the original inequality. Verification helps confirm that no errors occurred during the solution process.

In this example, after solving the inequality \(z < 5\), a test value, such as \(z = 4\), was substituted back into the original inequality \(\frac{z-2}{12}<\frac{1}{4}\) to check for correctness. The left side evaluated to \(\frac{1}{6}\), which is indeed less than \(\frac{1}{4}\), verifying that the solution is correct.

This verification step is crucial as it confirms that all transformations and operations applied during the solution process did not alter the inequality fundamentally, ensuring the correct set of solutions has been identified.