Problem 66 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ |3 x-5| \leq 2 $$

Step-by-Step Solution

Answer

The solution is \(1 \leq x \leq \frac{7}{3}\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|3x - 5| \leq 2\) involves an absolute value, which is defined as the distance from zero on the number line. Inequalities involving absolute values can be split into two separate inequalities.

Step 2: Remove Absolute Value and Split

Since the inequality is \(|3x - 5| \leq 2\), we can split it into two separate inequalities: 1. \(3x - 5 \leq 2\)2. \(3x - 5 \geq -2\).

Step 3: Solve the First Inequality

First, solve \(3x - 5 \leq 2\). Add 5 to both sides to isolate terms with \(x\): \(3x \leq 2 + 5\)\(3x \leq 7\). Now, divide by 3 to solve for \(x\): \(x \leq \frac{7}{3}\).

Step 4: Solve the Second Inequality

Next, solve \(3x - 5 \geq -2\). Add 5 to both sides: \(3x \geq -2 + 5\)\(3x \geq 3\). Divide by 3 to find \(x\): \(x \geq 1\).

Step 5: Combine the Solutions

Combine the results from both inequalities: \(1 \leq x \leq \frac{7}{3}\). This represents the solution set for the inequality \(|3x - 5| \leq 2\).

Key Concepts

Absolute Value

The absolute value of a number, denoted with the symbols \(|...|\), represents its distance from zero on the number line. This means the absolute value is always non-negative, regardless of whether the number inside the absolute value symbols is positive or negative. For example, \(|-3| = 3\) and \(|3| = 3\). Absolute value expressions become particularly interesting when they appear in inequalities. This is because they imply a range of values, rather than just one specific value. In the exercise \(|3x - 5| \leq 2\), this inequality suggests that the expression \(3x - 5\) must stay within two units of zero on the number line. These types of inequalities are solved by splitting them into two separate inequalities, capturing both the positive and negative range.

Solution Set

The solution set of an inequality is the collection of all possible values that satisfy the inequality condition. In the context of the absolute value inequality \(|3x - 5| \leq 2\), the solution set represents a range of values that satisfy both conditions derived from the inequality.

• The inequality \(3x - 5 \leq 2\) suggests a maximum boundary.
• The inequality \(3x - 5 \geq -2\) suggests a minimum boundary.

By solving both inequalities and finding values that meet both conditions, one can establish the solution set. This provides a complete picture of all possible solutions. For the exercise, the solution set \(1 \leq x \leq \frac{7}{3}\) represents a continuous range of values that satisfy \(|3x-5| \leq 2\). This means any value for \(x\) within this range will make the original inequality true.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or inequalities in order to find a solution. In our specific exercise, algebraic manipulation is used in solving both inequalities derived from the absolute value inequality.
Begin by removing the absolute value by setting up two inequalities: \(3x - 5 \leq 2\) and \(3x - 5 \geq -2\). This step effectively removes the absolute value barrier and translates it into algebraic expressions that can be solved conventionally.

• Add or subtract terms on both sides to isolate the variable term, \(3x\), which in our case involved adding five to simplify each inequality.
• Then, divide each side by the coefficient of \(x\) (which is three in this problem) to find \(x\). This manipulation results in: \(x \leq \frac{7}{3}\) and \(x \geq 1\).

It’s crucial for students to recognize how each manipulation helps converge toward a clear solution set. Mastery of these steps enables quick and reliable resolutions to similar problems.