Problem 112 - College Algebra

Question

When solving an inequality, when is it necessary to change the sense of the inequality? Give an example.

Step-by-Step Solution

Answer

When solving an inequality, the sense of the inequality needs to be changed whenever both sides of the inequality are multiplied or divided by a negative number. For instance, when solving -2x > 8, upon dividing both sides by -2 the solution becomes x < -4.

Step 1: Understand Inequalities

An inequality is an equation that uses any of the following signs: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), ≠ (not equal to). It is used to compare two values.

Step 2: Learn the Rule

In solving inequalities, the sense of the inequality is reversed, or changed, whenever both sides of the inequality are multiplied or divided by a negative number.

Step 3: Illustrate With an Example

Let's start with an inequality -2x > 8. To find the value of x we divide both sides by -2. According to the rule learned in step 2, we change the direction of the inequality sign, hence the solution will be x < -4. Therefore, for x values less than -4 the original inequality -2x > 8 holds true.

Key Concepts

Inequality Symbols

Inequalities are a way to express that one quantity is larger or smaller than another. Instead of using an equal sign, inequalities are represented with symbols such as:

• \( > \) : Greater than
• \( < \) : Less than
• \( \geq \) : Greater than or equal to
• \( \leq \) : Less than or equal to
• \( eq \) : Not equal to

These symbols help in comparing two values or expressions. For example, if you have \( x > 5 \), it means that \( x \) is greater than 5.

Reverse Direction

Changing the direction of an inequality symbol is crucial under specific conditions. This rule is unique to inequalities and does not apply to equalities.
Whenever you multiply or divide each side of an inequality by a negative number, the inequality symbol's direction must be reversed.This reversal reflects the change in order due to the effect of negative multiplication or division.
For example, if you have an inequality like \( -3x < 9 \), dividing by -3 to solve for \( x \) would necessitate flipping the symbol, resulting in \( x > -3 \). This ensures the inequality accurately reflects the relationship after the operation.

Multiply or Divide by Negative Number

Multiplying or dividing both sides of an inequality by a negative number may seem simple, but it carries a special rule that needs careful attention.
When a negative number is involved:

• The inequality symbol needs to be flipped. This is unlike regular equations where actions do not alter equality.
• This maintains the logical relationship between the numbers involved after the operation.

Consider \( -2x > 6 \). If you divide by \(-2\), you must flip the symbol, giving \( x < -3 \).This reflects that as \( x \) becomes less negative (larger numerically), it still satisfies the original condition.

Solve Inequalities

Solving inequalities involves finding the range of values that make the inequality true.
Here’s how you can solve them step by step:

• Isolate the variable you are solving for, just like you would in an equation.
• If you need to multiply or divide by a negative number, remember to reverse the inequality symbol.
• Verify by checking a few values from the solution against the original inequality.

Consider \( 3x + 5 \leq 11 \):

• Subtract 5: \( 3x \leq 6 \)
• Divide by 3: \( x \leq 2 \)

This means any value of \( x \) less than or equal to 2 will satisfy the inequality.