Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |3 x-7| \geq 5 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, \frac{2}{3}] \cup [4, \infty)\).

Step 1: Understanding Absolute Value Inequality

The inequality is \(|3x - 7| \geq 5\). This means that the expression inside the absolute value can either be greater than or equal to 5 or less than or equal to -5.

Step 2: Break Into Two Inequalities

We split \(|3x - 7| \geq 5\) into two separate inequalities: 1. \(3x - 7 \geq 5\) 2. \(3x - 7 \leq -5\). These handle the two scenarios for the absolute value.

Step 3: Solve First Inequality

Solve \(3x - 7 \geq 5\):Add 7 to both sides to get:\[3x \geq 12\]Then divide both sides by 3:\[x \geq 4\]

Step 4: Solve Second Inequality

Solve \(3x - 7 \leq -5\):Add 7 to both sides to get:\[3x \leq 2\]Then divide both sides by 3:\[x \leq \frac{2}{3}\]

Step 5: Combine Solutions

Since \(|3x - 7| \geq 5\), combine both parts of the solution. This represents values where the expression is either greater than or equal to 5 or less than or equal to -5.The solution in interval notation is: \((-\infty, \frac{2}{3}] \cup [4, \infty)\).

Key Concepts

Solving Inequalities

When dealing with inequalities, the goal is to find all values of the variable that satisfy the given condition. An absolute value inequality involves expressions with an absolute value that sets up conditions about how large or small the expression inside must be. Let's unpack this:

• The inequality \(|3x - 7| \geq 5\) means that the expression \(3x - 7\) is either greater than or equal to 5 or less than or equal to -5.
• To address this, we break it into two separate inequalities: \(3x - 7 \geq 5\) and \(3x - 7 \leq -5\).

By solving these two inequalities independently, we find the solution set for the entire inequality. This process includes adding or subtracting terms and then dividing by coefficients. The result is a range of values for the variable, which can be expressed using interval notation.

Interval Notation

To express solution sets, we often use interval notation as a way to describe parts of the number line where solutions can be found. Intervals are written in the form of (lower bound, upper bound), with different brackets to indicate whether the endpoints are included:

• Round brackets \(()\) mean that the endpoint is not included.
• Square brackets \([]\) mean the endpoint is included.

For our inequality \(|3x - 7| \geq 5\), the solution includes all numbers less than or equal to \(\frac{2}{3}\) and all numbers greater than or equal to 4. Hence, the solution is written in interval notation as \(( -\infty, \frac{2}{3}] \cup [4, \infty)\). The union symbol \(\cup\) indicates that these are separate intervals, combined to represent all possible solutions.

Algebraic Expressions

Algebraic expressions involve numbers and variables combined using arithmetic operations like addition, subtraction, multiplication, and division. Solving an inequality with an absolute value involves manipulating an algebraic expression to find the range of values for the variable. Here's a break down:

• First, the expression inside the absolute value, such as \(3x - 7\), must be isolated to set up the inequality.
• Then, solve the resulting simple inequalities (e.g., \(3x \geq 12\) or \(3x \leq 2\)) using basic algebra steps.
• These steps include operations such as adding or subtracting the same number from both sides and dividing both sides by the coefficient of the variable.

Understanding the structure and manipulation of algebraic expressions is crucial to solving inequalities and finding solutions that satisfy the original condition.