Problem 88 - Beginning and Intermediate Algebra

Question

Rewrite the following inequalities so that the inequality symbol points in the opposite direction and the resulting statement has the same meaning as the given one. $$ -4<-2 $$

Step-by-Step Solution

Answer

Change to \(-2 > -4\).

Step 1: Understand the Original Inequality

The original inequality is \(-4 < -2\), which means that -4 is less than -2. In simpler terms, -4 is to the left of -2 on a number line.

Step 2: Reverse the Inequality Symbol

The task requires reversing the inequality symbol. Turning \(<\) into \(>\) changes the statement to \(-4 > -2\). This is not correct with the same terms but helps understand the opposite direction.

Step 3: Swap the Sides to Maintain Meaning

To maintain the same meaning while using the reversed inequality symbol, swap the terms around the inequality symbol. This results in \(-2 > -4\), which states that -2 is greater than -4.

Step 4: Verify the Resulting Inequality

Check that the new inequality \(-2 > -4\) has the same logical meaning as the original inequality \(-4 < -2\). Indeed, -2 is greater than -4, reinforcing the correct transformation.

Key Concepts

inequalities reversal

Reversing inequalities is a crucial concept when solving algebraic problems. This operation involves changing the direction of the inequality symbol while keeping the inequality's true meaning intact. Here's how to do it correctly.

When you reverse an inequality like \(-4 < -2\), the inequality symbol changes from \(<\) to \(>\). This reversal does not work if you leave the numbers in the same positions because \(-4 > -2\) would be incorrect.Instead, you must also switch the positions of the numbers. This means the statement should be rewritten as \(-2 > -4\), which is logically true. By reversing the inequality symbol and swapping the numbers, you maintain the correct relationship between them.

number line positioning

Understanding number line positioning is essential for grasping inequalities. A number line visually represents numbers in increasing order from left to right. This visual tool helps in comparing numbers easily.

Take the inequality \(-4 < -2\). On the number line, \(-4\) appears to the left of \(-2\), confirming that it is smaller. The position helps us comprehend why this inequality is true.

By placing numbers on the number line, we validate the inequality. Positioning shows that leftmost values are smaller than those to the right. This reinforces the importance of number line positioning in understanding and working with inequalities.

inequality symbols

Inequality symbols are signs used in mathematics to compare the relative size of numbers or expressions. They communicate whether a number is greater than, less than, or equal to another number.

Common inequality symbols include:

• \(<\) : less than
• \(>\) : greater than
• \(\leq\) : less than or equal to
• \(\geq\) : greater than or equal to

When altering an inequality's symbol to its opposite, such as switching \(<\) to \(>\), it's vital to also rearrange the numbers or expressions involved. This adjustment ensures the inequality's meaning remains consistent.

By understanding how to appropriately use and reverse inequality symbols, you can better solve inequalities and accurately interpret mathematical relationships.