Problem 35 - Precalculus Mathematics for Calculus

Question

State whether each inequality is true or false. (a) \(-\pi>-3\) (b) \(8 \leq 9\)

Step-by-Step Solution

Answer

(a) False (b) True

Step 1: Understanding Inequalities

Before addressing each inequality, let's remind ourselves of the inequality symbols. The '>' symbol means greater than, whereas '≤' means less than or equal to.

Step 2: Evaluating (a) \(-\pi > -3\)

First, recognize the value of \(-\pi\), which is about -3.14. We compare this to -3. Since -3.14 is less than -3, the inequality \(-\pi > -3\) is false.

Step 3: Evaluating (b) \(8 \leq 9\)

The inequality symbol '≤' means 'less than or equal to'. Since 8 is certainly less than 9, the statement \(8 \leq 9\) is true.

Key Concepts

Inequality Symbols

Inequality symbols are essential tools in mathematics used to compare the sizes of numbers or expressions. When we talk about inequalities, we usually refer to symbols like:

• 'Greater than (>): This symbol indicates that the value on the left is larger than the value on the right. For example, in 5 > 3, 5 is greater than 3.
• 'Less than (<): This symbol means the value on the left is smaller than the value on the right. So, 2 < 4 indicates that 2 is less than 4.
• 'Less than or equal to (≤): This conveys that the value on the left is either smaller than or exactly equal to the value on the right. Thus, 2 ≤ 2 and 1 ≤ 4 both hold true.
• 'Greater than or equal to (≥): This tells us that the left side is either greater or equal to the right side.

Recognizing and understanding these symbols enable us to solve problems involving the comparison of numbers effectively.

Greater Than

The 'greater than' symbol, represented as '>', is used when we want to say that a number is larger than another. For example, if Alice has more apples than Bob, we can mathematically say: Number of Alice's apples > Number of Bob's apples. This symbol helps us make clear comparisons in mathematical problems and real-world situations.

When using 'greater than' in inequalities, the number or expression on the left side must indeed be larger than the one on the right. If we take the expression \(-3 > -4\), it holds because -3 is indeed greater than -4 on a number line. It's always important to visualize or think about number lines to better grasp these concepts.

Less Than or Equal To

The 'less than or equal to' symbol (≤) is a versatile tool in mathematics. It signals that a number is either smaller than or exactly the same as another number.
Let's use an everyday example: If a bus can carry a maximum of 30 passengers, the number of current passengers can either be less than 30 or just 30, because the maximum capacity is the limiting value. Here, the relationship can be written as Passengers ≤ 30.
In our mathematical exercise, when evaluating whether 8 ≤ 9, we find this statement true because 8 is indeed less than 9. This example highlights how useful and straightforward the ≤ symbol is in expressing constraints or relationships where values cannot exceed a certain point.

Evaluating Inequalities

Evaluating inequalities involves determining whether the statement made by an inequality symbol is true or false.

• To start evaluating, ensure you understand each symbol and its meaning. Knowing that '>' indicates larger and '≤' represents less than or equal to is crucial for this process.
• Take \(-\pi > -3\), where \( -\pi \) is approximately -3.14. With this in mind, when we compare -3.14 and -3, we recognize -3.14 is less than -3, making this inequality false.
• Another example is 8 ≤ 9. Here, you evaluate whether 8 is less than or equal to 9. As 8 is indeed less than 9, this inequality is true.

Understanding the principles behind inequalities not only helps solve problems but also assists in verifying solutions by reflecting on each comparison logically.