Problem 14 - Elementary and Intermediate Algebra

Question

Perform the necessary steps to isolate the absolute value expression on one side of the equation. Do not solve. a. \(|3 x+2|-7=-5\) b. \(6+2|5 x-19| \leq 40\)

Step-by-Step Solution

Answer

a. Isolated expression: \\( |3x + 2| = 2 \\\, b. Isolated expression: \\( |5x - 19| \\leq 17 \\\).

Step 1: Understanding the Expression (a)

Identify the structure of the expression \( |3x + 2| - 7 = -5 \). The goal is to isolate the absolute value expression \( |3x + 2| \) on one side of the equation. Currently, there is a \(-7\) that needs to be removed.

Step 2: Isolate |3x + 2| (a)

Add 7 to both sides of the equation to eliminate the \(-7\) from the left side: \( |3x + 2| - 7 + 7 = -5 + 7 \). This simplifies to \( |3x + 2| = 2 \).

Step 3: Understanding the Expression (b)

Identify the structure of the expression \( 6 + 2|5x - 19| \leq 40 \). Here, we need to isolate \( |5x - 19| \). The immediate step is to move the 6 on the left to the other side.

Step 4: Simplify the Inequality (b)

Subtract 6 from both sides to begin isolating the absolute value: \( 6 + 2|5x - 19| - 6 \leq 40 - 6 \). This results in \( 2|5x - 19| \leq 34 \).

Step 5: Isolate |5x - 19| (b)

Divide both sides by 2 to complete the isolation of the absolute value: \( \frac{2|5x - 19|}{2} \leq \frac{34}{2} \). This simplifies to \( |5x - 19| \leq 17 \).

Key Concepts

Isolation

When working with absolute value equations and inequalities, one of the key steps is to isolate the absolute value term. This makes it simpler to solve or further analyze the equation. The process often involves rearranging the equation by performing basic algebraic operations like addition, subtraction, multiplication, or division.

• For example, in the equation \(|3x + 2| - 7 = -5\), we need to eliminate the \(-7\) to make it easier to work with \(|3x + 2|\).
• This is achieved by adding 7 to both sides of the equation, resulting in \(|3x + 2| = 2\).
• Similar logic applies to inequalities; for \(6 + 2|5x - 19| \leq 40\), you would first subtract 6 and then divide by 2 to isolate \(|5x - 19|\).

Isolating the absolute value expression helps to focus on the essence of the problem, allowing you to apply further methods and solve for the variable efficiently.

Inequalities

Inequalities, much like equations, are mathematical statements that express a relationship between two expressions. However, instead of equating them, inequalities show how one expression is greater, lesser, or equal to another.
In the context of absolute value inequalities, the goal is to isolate the absolute value term in order to evaluate the range of possible solutions.Once isolated, you interpret the inequality into two separate cases since absolute values can represent both positive and negative scenarios.

• Using the example \(2|5x - 19| \leq 34\), the isolated inequality is \(|5x - 19| \leq 17\).
• This can be split into two inequalities: \(5x - 19 \leq 17\) and \(5x - 19 \geq -17\).

These two inequalities allow us to find the range of values for \(x\) that will satisfy the original inequality, considering both the positive and negative interpretations of the absolute value.

Algebraic Expressions

When dealing with algebraic expressions, whether they include absolute values or not, understanding the basic components and how to manipulate them is crucial. An algebraic expression consists of variables, numbers, and operations (addition, subtraction, multiplication, division).
Absolute value expressions are a special kind of algebraic expression that represent the distance of a number from zero on the number line, always being non-negative.

• For instance, \(|3x + 2|\) is an algebraic expression where \(3x + 2\) is the variable part inside the absolute value.
• In the equation \(|3x + 2| = 2\), manipulating the expression correctly while respecting the properties of absolute values helps in solving for \(x\).

By mastering the manipulation of algebraic expressions, you can break down complex problems into simpler parts, solve them efficiently, and gain a deeper understanding of the underlying mathematical relationships.