Problem 73 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |3 x-8|>7 $$

Step-by-Step Solution

Answer

The solutions for the inequality are \(x<1/3\) or \(x>5\).

Step 1: Remove the Absolute Value

To remove the absolute value, we need to separate the problem into two inequalities: \(3x-8>7\) and \(3x-8<-7\).

Step 2: Solve the First Inequality

The first inequality, \(3x-8>7\), simplifies to \(3x > 15\), and then to \(x > 5\), by adding 8 to both sides and then dividing by 3.

Step 3: Solve the Second Inequality

The second inequality, \(3x-8<-7\), simplifies to \(3x < 1\), and then to \(x < 1/3\), by adding 8 to both sides and then dividing by 3.

Step 4: Examine the Solutions

From Step 2, we find that \(x > 5\). From Step 3, we find that \(x<1/3\). Combining these we get either \(x<1/3\) or \(x>5\).

Key Concepts

Understanding Inequalities

Inequalities are comparisons between mathematical expressions that involve symbols like ">" (greater than), "<" (less than), "≥" (greater than or equal to), or "≤" (less than or equal to). They are important because they allow us to express ranges of possible solutions rather than a single value.
For example, the inequality \( x > 7 \) tells us that \( x \) can be any number greater than 7.
Inequalities can be used to define limits and give solutions in terms of intervals.Absolute value inequalities have an extra step because of the absolute value bars, which make the expression inside always non-negative. This means that we need to consider both the positive and negative cases for the term inside the absolute value.

Basic Concepts of Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It allows for the representation of relationships and patterns in numbers.
In our context with inequalities, algebra helps us manipulate terms and expressions to isolate the variable, which is the key to solving these inequalities.
Some basic operations involve:

• Adding or subtracting the same number on both sides of an equation or inequality.
• Multiplying or dividing both sides by a positive number.
• Reversing the inequality symbol if multiplying or dividing both sides by a negative number.

This manipulation is central to solving problems, as it allows for the transformation of an expression into a more usable form.

Steps to Solving Inequalities

The process of solving inequalities often involves similar steps to solving equations, but with important distinctions. When dealing with absolute value inequalities, the problem usually requires solving two separate inequalities.
For instance, in an inequality like \(|3x-8|>7\), we split it into two separate inequalities: one for the positive case \(3x-8>7\) and one for the negative case \(3x-8<-7\).
Here's a simple step-by-step approach:

• Start by removing the absolute value by setting up two inequalities.
• Solve each inequality separately by isolating the variable using algebraic operations.
• Combine the solutions to understand the range of values that satisfy the original inequality.

When solving, remember to check if any steps require reversing the inequality sign, particularly when dividing by a negative number.

Crafting Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols. In inequalities, expressions can define conditions that must be met for a solution.
Crafting expressions requires precision and understanding of algebraic principles.
Each part of an expression must be clear:

• Numbers and constants are straightforward but they set the scale or size of the equation.
• Variables act as placeholders and allow for a wide range of values within expressions.
• Operations determine how numbers are calculated and include addition, subtraction, multiplication, and division.

A good understanding of how expressions work helps in setting up and manipulating inequalities to find solutions. This brings clarity to problems involving ranges and limits, like determining specific conditions for solutions in inequalities.