Problem 41 - Algebra for College Students

Question

Solve each equation and inequality. \(\left|\frac{x-3}{4}\right|<2\)

Step-by-Step Solution

Answer

The solution is \(-5 < x < 11\).

Step 1: Understand the Absolute Value Inequality

The given inequality is \( \left|\frac{x-3}{4}\right|<2 \). This reads as the absolute value of the expression \( \frac{x-3}{4} \) being less than 2. We need to split this into two separate inequalities to remove the absolute value.

Step 2: Remove the Absolute Value

For any expression \( |A| < B \), it implies \(-B < A < B\). Thus, for \( \left|\frac{x-3}{4}\right|<2 \), we write the inequalities \(-2 < \frac{x-3}{4} < 2\).

Step 3: Solve the Left Inequality

For \(-2 < \frac{x-3}{4}\):1. Multiply each term by 4 to eliminate the fraction: \(-2 \cdot 4 < x-3\).2. Simplify to get: \(-8 < x-3\).3. Add 3 to each side of the inequality: \(-8 + 3 < x\).4. Thus, \(-5 < x\).

Step 4: Solve the Right Inequality

For \(\frac{x-3}{4} < 2\):1. Multiply each term by 4: \( x-3 < 2 \cdot 4 \).2. Simplify to get: \( x-3 < 8 \).3. Add 3 to each side of the inequality: \( x < 8 + 3 \).4. Thus, \( x < 11 \).

Step 5: Combine the Results

Combine the results of steps 3 and 4. The solution to the inequality is the intersection of the two conditions: \(-5 < x < 11\).

Key Concepts

Solving Inequalities

Inequalities are statements that relate algebraic expressions using the inequality signs such as `<`, `>`, `≤`, or `≥`. When solving inequalities, our aim is to find the range of values that satisfy the inequality. With absolute value inequalities, this process involves a few extra steps.

To begin with, understanding the "absolute value" is crucial. Absolute value refers to a number's distance from zero on the number line, without considering direction. So, an absolute value inequality such as \(|A| < B\) means the absolute value of A is less than B. Content-wise, this translates to two separate inequalities: \(-B < A < B\).

Let's apply this to our example, \(\left|\frac{x-3}{4}\right|<2\). Here, we split it into two parts: \-2 < \frac{x-3}{4} < 2\. This splitting is the key first step in solving absolute value inequalities. Once split, you can solve each inequality separately, much like you would handle regular linear inequalities. This helps you narrow down the solution to the range of x that works for both inequalities.

Inequality Solutions

The solutions to inequalities are often ranges of values rather than single numbers, as is often the case with equations. For the inequality \-5 < x < 11\, the solution set means x can be any number greater than -5 but less than 11.

To solve these inequalities, you generally isolate the variable (in this case, x) by performing operations such as addition, subtraction, multiplication, or division. However, it's important to remember that when you multiply or divide by a negative number, the inequality sign reverses.

In our original solution, the inequality \(-2 < \frac{x-3}{4} < 2\) was converted into \-8 < x-3\ and \x-3 < 8\. Through these steps, we simply worked ladder-like with both inequalities until x was isolated. Always ensure both parts of an inequality are true for the solution to be valid in joint terms.

Algebraic Expressions

In algebra, expressions are combinations of numbers, variables, and operators (such as + and -). In dealing with inequalities that involve algebraic expressions, it's important to first simplify the expressions as much as possible.

Consider the fraction in the expression \(\frac{x-3}{4}\) within our exercise. The goal was to eliminate the fraction to simplify the process of isolating x. This involves multiplying the entire inequality by 4, the denominator. Doing so transforms the expressions: you end up with an expression without fractions, which are easier to manipulate.

• You can deal with two inequalities separately.
• Keep the solution steps consistent, performing the same operations on each part.

Fractions, distributed terms, or grouped expressions, can make problems complex. Simplifying wherever possible prevents errors and keeps the steps direct and clear.