Question

State whether each of the following statements is true or false. \(\begin{array}{lll}\text { (a) } 4>9 & \text { (b) } 4>4 & \text { (c) } 4 \geq 4\end{array}\) (d) \(0.001<10^{-5}\) (e) \(|-19|<100\) (f) \(|-19|>-20\) (g) \(0.001 \leq 10^{-3}\)

Short answer

Question: Determine whether the following statements are true or false. a) 4 > 9 b) 4 > 4 c) 4 ≥ 4 d) 0.001 < 10^(-5) e) |-19| < 100 f) |-19| > -20 g) 0.001 ≤ 10^(-3) Answer: a) False b) False c) True d) False e) True f) True g) True

Step-by-step solution

Statement (a): 4 > 9

Compare the numbers 4 and 9. Since 4 is less than 9, the inequality 4 > 9 is false.

Statement (b): 4 > 4

Compare the number 4 to itself. Since 4 is equal to 4, the inequality 4 > 4 is false.

Statement (c): 4 ≥ 4

Compare the number 4 to itself. Since 4 is equal to 4, the inequality 4 ≥ 4 is true.

Statement (d): 0.001 < 10^(-5)

Convert 0.001 to scientific notation: \(1 \times 10^{-3}\). Compare this to \(10^{-5}\). Since -3 is greater than -5, the inequality 0.001 < 10^(-5) is false.

Statement (e): |-19| < 100

Evaluate the absolute value of -19, which is 19. Compare this to 100. Since 19 is less than 100, the inequality |-19| < 100 is true.

Statement (f): |-19| > -20

Evaluate the absolute value of -19, which is 19. Compare this to -20. Since 19 is greater than -20, the inequality |-19| > -20 is true.

Statement (g): 0.001 ≤ 10^(-3)

Convert 0.001 to scientific notation: \(1 \times 10^{-3}\). Compare this to \(10^{-3}\). Since both values are equal, the inequality 0.001 ≤ 10^(-3) is true.

Key concepts

Understanding Absolute Value

Absolute value is a way of describing how far a number is from zero on the number line, regardless of its direction. The absolute value of a number is always non-negative. For instance, the absolute value of both -19 and 19 is 19. This helps simplify comparisons:

• Expression: |-19| = 19
• This shows the distance from zero is 19, ignoring the negative sign.

For the comparison \(|-19| < 100\), we see that 19 is indeed less than 100, making the statement true. Similarly, \(|-19| > -20\) is true because 19 is greater than -20.
Understanding absolute value simplifies the process of comparing numbers, especially when negatives are involved.

Using Scientific Notation

Scientific notation is a helpful way to express very large or very small numbers, using powers of 10. This makes it easier to compare such numbers by looking at the exponents:

• 0.001 in scientific notation: \(1 \times 10^{-3}\)
• Comparing with: \(10^{-5}\)

When comparing the exponent values, -3 is greater than -5. So, 0.001 is actually greater than \(10^{-5}\), making the statement \(0.001 < 10^{-5}\) false. On the other hand, when comparing \(0.001 \leq 10^{-3}\), both are equal, making this statement true. Scientific notation provides a clear method for comparing the magnitude of numbers effectively.

Comparison of Real Numbers

Comparing real numbers is the fundamental skill of identifying which number is larger, smaller, or equal. The key is to line up the numbers or their values clearly:

• 4 vs 9: 4 is less than 9, so 4 > 9 is false.
• 4 vs 4: 4 is equal to 4, so 4 > 4 is false, but 4 \(\geq\) 4 is true.

By clearly evaluating each pair of numbers, we determine which comparison symbols to use. Understanding this comparison process allows better analysis of inequalities and proper application of greater than, less than, and equal to symbols in mathematical expressions.