Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each inequality. $$7 x \leq 7(x-2)$$

Step-by-Step Solution

Answer

The given inequality has no solution when approached in a linear fashion, however considering the property of inequality, all real values of x satisfy the given inequality. Therefore, the solution is \( x \in (-\infty, +\infty) \)

Step 1: Simplify the inequality

First, distribute the 7 on the right-hand side of the inequality: \(7x \leq 7x - 14\)

Step 2: Isolate x

Subtract \(7x\) from both sides of the inequality to isolate \(x\) on one side: \(0 \leq -14\)

Step 3: Check for solutions

The inequality \(0 \leq -14\) has no solution, because 0 is not smaller or equal to -14. However, it must be noted that initial subtraction of \(7x\) from both sides resulted in losing possible solutions, as this manipulation doesn't consider the change of direction in inequality when multiplying by -1

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It allows us to simplify expressions by distributing a multiplication across terms in parentheses. In this exercise, we're applying the distributive property to open up the expression on the right side of the inequality.

• This means multiplying the 7 by each term inside the parentheses: \[ 7(x - 2) = 7 \times x - 7 \times 2 = 7x - 14 \]
• By distributing, we effectively remove the parentheses, which makes it easier to further manipulate and solve the inequality.

Understanding the distributive property is key for many algebraic processes, particularly when working with complex equations or inequalities. By mastering this rule, you can simplify tasks and make solving inequalities more straightforward.

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation or inequality. This is crucial for finding the solution to inequalities, as it reveals the conditions under which the variable satisfies the inequality.

• In our exercise, after distributing, we started with: \[ 7x \leq 7x - 14 \]
• To isolate \(x\), we attempted to subtract \(7x\) from both sides, which resulted in: \[ 0 \leq -14 \]
• This reveals a fundamental aspect of isolating: sometimes transformations can lead to surprising results, such as here, indicating there are no solutions.

Always be cautious when manipulating inequalities. If you multiply or divide by a negative number, remember to flip the inequality sign. This is often where mistakes can be made when isolating variables in inequality.

Solution Check

Checking your solution is an important step in solving inequalities. It ensures that the conclusion you've reached makes logical sense and aligns with the mathematical rules applied.

• In this problem, the end result \(0 \leq -14\) seemed incorrect. Generally, this type of contradiction indicates that there might be no solution or all real numbers are solutions in cases where variables cancel completely and lead to true statements (like \(0 = 0\)).
• The error lies in the process of solving the inequality—often due to overlooking steps like the impact of subtracting similar terms.
• By verifying against the original inequality and considering the validity of each manipulation step, it's crucial to validate initial assumptions and transformations.

Validating solutions is a great habit, as it sharpens accuracy and comprehension. Always reassess your steps in algebra to confirm their correctness.