Problem 148 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,1\) can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

Step-by-Step Solution

Answer

The statement does not make sense because the decision to divide or multiply by a negative is determined by the coefficients of the variable terms in the inequality, and not by the position of the constant terms. This rule always applies when manipulating inequalities.

Step 1: Understanding the property

Let's recall a critical property of inequalities, which states that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if \(-2x > 6\), to isolate \(x\), we need to divide both sides by \(-2\) which gives \(x < -3\). This is a fundamental property of inequalities that holds true no matter how the components are arranged.

Step 2: Refuting the statement

Now, let's look at the given inequality \(5x + 4 < 8x - 5\). No matter how we shuffle the terms around - say, moving \(5x\) to the right and \(-5\) to the left, or \(8x\) to the left and \(4\) to the right - there is no way we can avoid the rule of changing the sign of the inequality when we divide by a negative number. This is because the decision to divide by a negative number depends on the coefficients of \(x\), not on where the constant terms are positioned.

Key Concepts

Dividing Inequalities

When working with inequalities, division is a crucial operation that can affect the direction of the inequality. When you divide both sides of an inequality by a positive number, the inequality sign remains the same. However, dividing by a negative number adds an extra step.
This is because dividing by a negative number flips the inequality sign. For example, if you have the inequality \(-2x > 6\), dividing both sides by \(-2\) gives us \(x < -3\).
Remember:

• Dividing by positive? Keep the sign.
• Dividing by negative? Flip the sign!

Being cautious about this sign change is critical to solving inequalities correctly.

Reversing Inequality Sign

One unique property of inequalities is sign reversal. This occurs during multiplication or division by a negative number.
Consider this: when we multiply or divide by negative numbers, we must change the direction of the inequality. This means that \(>\) becomes \(<\) and vice versa.
Why does this happen? It's a direct result of how numbers are ordered on a number line. For instance, \(-3 > -4\) when multiplied by \(-1\) becomes \(3 < 4\).

• Always check the number you're using to multiply or divide.
• Be aware of negative numbers to avoid mistakes in direction.

Understanding this concept is essential for working confidently with inequalities.

Linear Inequalities

Linear inequalities are similar to linear equations but with inequality signs (\(<, >, \leq, \geq\)). Solving them involves similar steps to solving linear equations: isolating the variable on one side.
Consider the inequality \(5x + 4 < 8x - 5\). The goal is to get \(x\) alone. First, we consolidate terms involving \(x\) on one side and constant terms on the other.
This process might involve moving terms around – for instance:

• Subtract \(5x\) from both sides: \(4 < 3x - 5\).
• Add \(5\) to both sides: \(9 < 3x\).

Straightforward and efficient steps like these are key to solving linear inequalities.

Solving Inequalities

Solving inequalities follows a structured approach, allowing us to find the range of possible solutions for a variable. Here's how you can tackle problems like \(5x + 4 < 8x - 5\):
1. **Move Variables and Constants:** Arrange variables on one side and constants on the other to simplify the expression.
2. **Isolate the Variable:** When moving terms, remember to perform operations on both sides to keep the inequality balanced.
3. **Check Negative Coefficients:** If you divide or multiply by a negative, remember to flip the inequality sign.
These steps help identify the solution set where the inequality holds true. Practice makes it easier to navigate these processes efficiently.