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Chapter 2: Problem 85

When solving an inequality, when must you reverse the direction of the inequality symbol?

Short Answer

Expert verified
Reverse the inequality symbol when multiplying or dividing by a negative number.

Step by step solution

01

Understanding Inequalities

An inequality is a mathematical statement that shows the relationship between two expressions, using symbols such as >, <, 鈮�, or 鈮�. For example, x > 5 means that x is greater than 5.
02

When to Reverse the Inequality Symbol

The inequality symbol must be reversed when multiplying or dividing both sides of the inequality by a negative number. For instance, if you have -2x > 6 and you divide by -2, you reverse the sign, resulting in x < -3.
03

Why Reverse the Symbol?

Reversing the inequality symbol is necessary because multiplying or dividing by a negative number changes the order of the values. This ensures that the inequality remains accurate and valid.

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Key Concepts

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

Inequality Symbols
Inequality symbols are essential in mathematics as they help us compare two different values or expressions. Each symbol has a specific meaning:
  • Greater than (>): This symbol indicates that the value on the left is larger than the value on the right. For example, in the inequality \( x > 3 \), \( x \) represents a number greater than 3.
  • Less than (<): This shows the opposite, where the left value is smaller. For example, \( y < 10 \) means \( y \) is a number smaller than 10.
  • Greater than or equal to (鈮�): This symbol means the left side is larger or equal to the right. For instance, \( a \geq 5 \) includes all numbers greater than or equal to 5.
  • Less than or equal to (鈮�): It illustrates that the left side is smaller or equal. For example, \( b \leq 8 \) means \( b \) is less than or equal to 8.
Understanding these symbols is crucial when solving inequalities. They tell us how expressions relate to each other, guiding us in finding solutions.
Multiplying and Dividing Inequalities
When working with inequalities, multiplying and dividing are operations that require careful attention. Here's why:
  • Regular Multiplication and Division: If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains unchanged. For example, if \( x < 5 \), then multiplying both sides by 2 results in \( 2x < 10 \).
  • Impacts of Negative Numbers: The story changes when the numbers are negative. Multiply or divide both sides of an inequality by a negative number, and you must reverse the inequality direction. This happens because the relative order of the numbers flips when negatives are introduced. For instance, if you start with \( -3y > 9 \), dividing by \(-3\) yields the inequality \( y < -3 \).
Let's think of this in terms of temperature shifting from cold to hot through zero. When the thermometer flips from negative to positive or vice versa, it alters the comparative scales, similar to reversing inequalities.
Reversing Inequality Direction
Reversing the inequality direction isn't just a random rule; it's a logical step that ensures the math stays accurate. Imagine comparing two temperatures: saying today is warmer than yesterday (10掳C > 0掳C). But if both days were compared using Fahrenheit with negatives (-10掳F > -5掳F is false; actually -10掳F < -5掳F), you need to adjust your comparison. The need to reverse comes into sharp focus when:
  • Multiplying/Dividing by Negatives: Whenever a negative number enters through multiplication or division, reversing maintains the inequality. Without reversing, you'd incorrectly interpret the relationship.
  • Conceptual Understanding: Think of a number line. Multiplying by a negative spins around your viewpoint, shifting greater numbers to lesser positions. This reversal of perspective is mirrored in the inequality requiring a flip.
This rule is why reversing the inequality direction in such situations is crucial, as it maintains a true representation of the initial mathematical statement.

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Most popular questions from this chapter

Dennis and Nancy Wood are celebrating their 30 th wedding anniversary by having a reception at Tiffany Oaks reception hall. They have budgeted \(\$ 3000\) for their reception. If the reception hall charges a \(\$ 50.00\) cleanup fee plus \(\$ 34\) per person, find the greatest number of people that they may invite and still stay within their budget.

Over the past few years the satellite Voyager II has passed by the planets Saturn, Uranus, and Neptune, continually updating information about these planets, including the number of moons for each. Uranus is now believed to have 13 more moons than Neptune. Also, Saturn is now believed to have 2 more than twice the number of moons of Neptune. If the total number of moons for these planets is 47 , find the number of moons for each planet. (Source: National Space Science Data Center)

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Solve each inequality. Write each answer using solution set notation. $$ 3(x+2)-6>-2(x-3)+14 $$

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