Problem 19 - Elementary Algebra Within Reach

Question

Match the statement with the property it represents. (a) Addition Property of Inequality (b) Subtraction Property of Inequality (c) Multiplication Property of Inequality (d) Division Property of Inequality \(3<9\), so \(3-1<9-1\).

Step-by-Step Solution

Answer

The property represented by the statement is (b) Subtraction Property of Inequality.

Step 1: Identify the operation

In the given inequality, the number '1' is subtracted from both '3' and '9'. This is a subtraction operation.

Step 2: Match with the property of inequality

The operation is subtraction, so the property in use here is the Subtraction Property of Inequality.

Key Concepts

Subtraction Property of Inequality

When you have an inequality and you subtract the same number from both sides, this is known as the Subtraction Property of Inequality. For example, given the inequality \( a < b \), if you subtract the same value, say \( c \), from both sides of the inequality, the result will be \( a-c < b-c \). This property is vital because it keeps the relationship between the two sides of the inequality unchanged.
What is important to understand here is that this property allows us to simplify inequalities and solve them much like we solve equations when the goal is to isolate a variable.

• Keep in mind: Subtract the same value from both sides.
• This process does not change the direction of the inequality.

This is why in the exercise, when 1 was subtracted from both 3 and 9, the inequality maintained its direction: \(3 - 1 < 9 - 1\). It's an essential tool for solving inequalities consistently.

Addition Property of Inequality

The Addition Property of Inequality allows you to add the same number to both sides of an inequality without changing the inequality’s direction. Suppose you have an inequality such as \( a < b \), adding the same number \( c \) to both sides gives you \( a+c < b+c \). This property helps in maintaining the truth of the inequality while allowing transformations that simplify or solve it.

• Use this property to make inequalities easier to handle.
• Maintain balance by adding the same number to both sides.

Just remember that the direction of the inequality remains unchanged when applying this property. It's a useful tool when you want to eliminate terms to isolate variables or rearrange terms while solving inequalities.

Multiplication Property of Inequality

Multiplying both sides of an inequality by a positive number is straightforward: the inequality's truth remains unchanged. Say you have \( a < b \) and you want to multiply by a positive number \( c \), the new inequality would be \( ac < bc \). This property is pretty handy when scaling numbers in inequalities.
However, if you multiply both sides of an inequality by a negative number, the inequality sign must flip. So \( a < b \) becomes \( ac > bc \) if \( c \) is negative. This flipping rule is crucial because it accounts for the direction of the number line.

• Multiply by positive numbers without flipping the sign.
• Multiply by negative numbers and remember to flip the sign.

Handling this property correctly helps to maintain the inequality’s truth while performing multiplicative operations.

Division Property of Inequality

Similar to multiplication, the Division Property of Inequality involves dividing both sides of an inequality by the same number. If the number is positive, the direction of the inequality doesn't change. For \( a < b \) and a positive number \( c \), the inequality \( \frac{a}{c} < \frac{b}{c} \) holds true.
Yet, if you divide by a negative number, the inequality sign flips. This means for \( a < b \) and a negative \( c \), the inequality transforms into \( \frac{a}{c} > \frac{b}{c} \). Always be cautious: the sign flips only when dividing by negatives due to their directional effect on the value relationship.

• Dividing by positive: no sign change necessary.
• Dividing by negative: always flip the sign.

Being vigilant with sign changes during division keeps your inequalities correct through transformations.