Problem 82 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. $$2 x-5 \geq 2 x$$

Step-by-Step Solution

Answer

The statement \(2x-5 \geq 2x\) is false.

Step 1: Analyzing the inequality

The given inequality is \(2x-5 \geq 2x\). To verify whether this inequality holds, manipulate the inequality to isolate the variable on one side.

Step 2: Simplify the inequality

Subtract \(2x\) from both sides in order to simplify the inequality. This results to the inequality: \(-5 \geq 0\)

Step 3: Evaluate

The inequality \(-5 \geq 0\) states that negative five is greater than or equal to zero. This is clearly false, which means that the initial inequality \(2x-5 \geq 2x\) is false.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves altering and rearranging algebraic expressions to make them more convenient to work with. It often includes operations such as addition, subtraction, multiplication, and division involving variables and constants. Our goal is to transform an expression or equation into a simpler or more useful form. For the problem given, we used subtraction as our method of manipulation.

First, we see the inequality:

• \(2x - 5 \geq 2x\)

To isolate the variable term, subtract \(2x\) from both sides. It's crucial that whatever you do to one side of the inequality, you must do to the other side as well. This keeps the inequality balanced.

The result is:

• \(-5 \geq 0\)

This process shows us that proper manipulation simplifies the complexity of the equations, allowing us to gain a clearer understanding and proceed with solution steps.

Solving Inequalities

Solving inequalities involves finding all the values for the variable that make the inequality true. Unlike equations, inequalities maintain a direction (\(>\), \(<\), \(\geq\), or \(\leq\)). Solving a linear inequality means finding a range of solutions that makes the inequality valid.

In this exercise, after performing algebraic manipulation, we derived the inequality:

• \(-5 \geq 0\)

Here, our task is to determine whether the resulting statement is true. It’s essential to pay close attention to the inequality symbol throughout the process. If we had multiplied or divided by a negative number, the direction of inequality would have flipped. But, here, no such action occurs, so it retains its direction.

Checking this statement against basic number understanding leads to the realization that solving inequalities isn't just about technical manipulation, but also about logical assessment.

Evaluating Statements

Evaluating mathematical statements means determining whether the statement is true or false. It involves critically analyzing the results of algebraic manipulations and comparing them with known truths.

In our given exercise, the statement we arrived at was:

• \(-5 \geq 0\)

Evaluating this is simple, as we're checking if \(-5\) is greater than or equal to \(0\). Clearly, \(-5\) is neither greater than nor equal to \(0\), so this is a false statement.

Understanding how to evaluate such resulting statements is essential because it is the final step that confirms whether the original problem conditions are met or not. Using logic and number sense helps corroborate that our previous algebraic operations were carried out correctly and assist in concluding our task effectively.