Problem 81 - Introductory Algebra for College Students

Question

Solve each inequality. \(4 x-4<4(x-5)\)

Step-by-Step Solution

Answer

The inequality has no solution

Step 1: Distributing on the right side

We distribute 4 to \(x - 5\) to get \(4*x - 4*5\). The inequality now looks like this: \(4x - 4 < 4x - 20\)

Step 2: Combining similar terms

We subtract \(4x\) from both sides of the inequality to balance it. We then get \(-4 < -20\)

Step 3: Interpreting the inequality

This shows an inconsistency because -4 is not less than -20. Therefore, the inequality has no solution

Key Concepts

Distributive Property

The distributive property is an essential tool when working with inequalities and equations. It allows us to simplify expressions by multiplying a single term across terms within parentheses. This property is especially useful for eliminating parentheses and clearing up complex expressions. In this exercise, the distributive property helps to simplify the equation on the right side.

- Initial equation: \(4x - 4 < 4(x - 5)\)
- Applying the distributive property: \(4 \cdot x - 4 \cdot 5\)
- Simplified result: \(4x - 4 < 4x - 20\)

By distributing, we have rewritten the expression without parentheses, making it easier to analyze and manipulate further. This step is vital before moving on to other operations like adding or subtracting terms.

Inconsistencies in Inequalities

Inequalities can sometimes show inconsistencies. This means that the resulting inequality makes a statement that is mathematically impossible or false. When simplifying or solving an inequality, it's crucial to pay attention to such results.

In our exercise, after using the distributive property and simplifying, we reach: \(-4 < -20\). This is clearly false because -4 is greater than -20. Such a scenario indicates an inconsistency, signaling that the original inequality holds no real solutions.

When dealing with inconsistencies:

• Verify each step to ensure no calculation errors.
• Understand that the inequality might contradict itself due to the problem constraints.
• Conclude that there are no real solutions unless any assumptions change.

Recognizing inconsistencies is a key step in solving and interpreting inequalities accurately.

Subtraction in Inequalities

Subtraction in inequalities is used to maintain balance while isolating variables. When you subtract the same term from both sides of an inequality, the inequality's direction remains unchanged. This operation is crucial for simplifying and solving inequalities.

In the given problem, subtracting \(4x\) from both sides results in:
- Before subtraction: \(4x - 4 < 4x - 20\)
- After subtraction: \(-4 < -20\)

We utilize subtraction to eliminate terms, revealing the core inequality. However, this process might also indicate an inconsistency if the result does not logically hold, as seen here.

When subtracting in inequalities, remember:

• The inequality direction doesn't change with subtraction.
• Check the resulting expression for logical accuracy.
• Use subtraction in combination with other operations for effective solving.

By mastering subtraction in inequalities, one can fluidly simplify and solve inequalities with confidence.