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Chapter 2: Problem 510

Explain why it is necessary to reverse the inequality when solving \(-5 x>10\)

Short Answer

Expert verified
When dividing or multiplying by a negative number, the inequality sign must be reversed. The solution is \( x < -2 \).

Step by step solution

01

Understand the Problem

We need to solve the inequality \( -5x > 10 \) and explain why it is necessary to reverse the inequality.
02

Isolate the Variable

To isolate \(x\), divide both sides of the inequality by \(-5\). This gives us \(-5x / -5 > 10 / -5\).
03

Simplify Both Sides

Simplify both sides. This results in \(x < -2\). Notice that the inequality symbol has been reversed.
04

Explain the Reversal

When dividing or multiplying both sides of an inequality by a negative number, the inequality sign must be reversed to maintain the truth of the statement. This happens because multiplying by a negative number reverses the order relationship between the numbers.
05

Final Answer

The solution to the inequality \( -5x > 10 \) is \( x < -2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reversing inequality
When solving inequalities, it's essential to understand the concept of reversing the inequality sign. This occurs primarily when we multiply or divide both sides of an inequality by a negative number.

  • In inequalities, we are evaluating relationships between numbers. For example, if we say that 5 is greater than 3, written as 5 > 3.
  • However, when you multiply or divide by a negative number, the order of these relationships changes.
  • This is because negative numbers flip the direction of the inequality.

    • Imagine you have the inequality -5x > 10. To isolate x, you need to divide by -5. Since you are dividing by a negative number, you must reverse the inequality sign from > to <, making the result x < -2. Reversing the inequality keeps the relationship between numbers truthful.
isolating variable
Isolating the variable is a key step in solving inequalities. This process involves manipulating the inequality to get the variable by itself on one side of the inequality sign.

Here are the steps to isolate the variable:
  • Identify the variable you need to isolate. In the example -5x > 10, the variable to isolate is x.
  • Use algebraic operations to move other elements away from the variable. For the given inequality, divide both sides by -5.

    • This operation removes the -5 from the left side, giving you x on its own. It's crucial to perform the same operation on both sides of the inequality to keep it balanced. After dividing by -5, you get x < -2.
dividing by negative number
One specific rule when dealing with inequalities is what happens when you divide by a negative number.

Consider this:
  • Dividing by a positive number keeps the inequality direction the same.
  • But dividing by a negative number reverses the inequality direction.

Why does this happen?
  • Negative numbers change the relative positions of numbers on the number line. For instance, if positive numbers make larger numbers 'stay larger', negatives do the opposite since -3 is less than -1 even though 3 is greater than 1.
  • Therefore, when you divide by a negative, you switch the inequality to reflect the actual relationship between numbers.

For our example -5x > 10, dividing by -5 requires switching the inequality from > to < to get the right solution, x < -2.

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