Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$|4-3 x|>1$$

Step-by-Step Solution

Answer

\((-\infty, \frac{5}{3}) \cup (1, \infty)\)

Step 1: Understand the Absolute Value Inequality

The inequality \(|4 - 3x| > 1\) is an absolute value inequality. Absolute value inequalities can be split into two separate inequalities. This happens because we are measuring the distance from 4 - 3x to zero, which can be greater than 1 in two possible scenarios.

Step 2: Split the Inequality

Split the inequality \(|4 - 3x| > 1\) into two separate inequalities:1. \(4 - 3x > 1\)2. \(4 - 3x < -1\)

Step 3: Solve the First Inequality

Solve the inequality \(4 - 3x > 1\):\[4 - 3x > 1 \-3x > 1 - 4 \-3x > -3 \x < 1 \ \]Notice that the inequality sign flips when dividing by a negative number.

Step 4: Solve the Second Inequality

Solve the inequality \(4 - 3x < -1\):\[4 - 3x < -1 \-3x < -1 - 4 \-3x < -5 \x > \frac{5}{3} \ \]Again, note the sign change when dividing by a negative.

Step 5: Combine the Solutions

Combine the solutions from the two inequalities:\(x < 1\) and \(x > \frac{5}{3}\).Thus, the solution is the union of these intervals: \(x < 1\) or \(x > \frac{5}{3}\).

Step 6: Express the Solution

The solution in interval notation is: \((-\infty, \frac{5}{3}) \cup (1, \infty) \)This shows that any \(x\) in these intervals will satisfy the original inequality.

Key Concepts

Inequality Solving

Solving inequalities is a key skill in algebra that helps us understand the range of possible values for a variable. Unlike equations, where the goal is to find exact numbers that satisfy the problem, inequalities deal with a wide spectrum of solutions. In inequality solving, we tackle expressions like

• "less than" (\(<\))
• "greater than" (\(>\))

which forms the basis of determining solution boundaries.
When approaching an inequality such as \(|4 - 3x| > 1\), it is important to realize that there are often multiple solutions since it involves a range of values rather than a single answer. The process often involves breaking down the inequality into simpler components and handling each separately. This encourages flexibility in thinking, helping us explore all potential solutions that fit the given conditions.

Absolute Value

An absolute value represents the distance of a number from zero on the number line, without considering direction. It is denoted with vertical bars like this: \(|x|\). For example, both 3 and -3 have an absolute value of 3 because they are both three units away from zero.
When dealing with absolute value inequalities, such as \(|4 - 3x| > 1\), we must consider both the positive and negative scenarios that could make the inequality true. Hence, we split it into two separate cases:

• The expression inside the absolute value is larger than 1 (\(4 - 3x > 1\)).
• The expression inside is smaller than -1 (\(4 - 3x < -1\)).

This dual approach allows us to capture the full range of possible solutions, as absolute values account for both positive and negative orientations from zero.

Interval Notation

Interval notation is a mathematical shorthand for describing ranges of numbers, particularly useful for expressing solutions to inequalities. It uses parentheses and brackets to signify open or closed intervals, respectively.
In our exercise, after solving the inequalities, we obtained the solutions \(x < 1\) and \(x > \frac{5}{3}\). In interval notation, these solutions are written as \((-\infty, \frac{5}{3}) \cup (1, \infty)\), where:

• \((-\infty, \frac{5}{3})\) represents all values less than \(\frac{5}{3}\).
• \((1, \infty)\) represents all values greater than 1.

Open intervals denote values that do not include the endpoints themselves, and when solutions are joined with the "union" symbol \((\cup)\), it signifies that solutions can belong to either one interval or the other.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions and equations to solve problems. It involves basic operations such as addition, subtraction, multiplication, and division.
In our inequality solution process, various algebraic manipulations are used:

• Rearranging terms by adding or subtracting numbers from both sides.
• Dividing or multiplying both sides by a number. Important: Dividing/multiplying by a negative number flips the inequality sign.

For example, when solving the inequality \(4 - 3x > 1\), we subtracted 4 from both sides, then divided by -3, remembering to reverse the inequality, yielding \(x < 1\). These steps showcase the importance of methodically applying algebraic operations while keeping track of the rules, especially those specific to inequalities.