Problem 81 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(-|8|>|-9|\)

Step-by-Step Solution

Answer

True

Step 1: Evaluate the Absolute Value

First, find the absolute value of 8. The absolute value of 8 is 8 because it is already a positive number.

Step 2: Apply the Negative Sign

Apply the negative sign to the absolute value of 8. This gives \(-|8| = -8\).

Step 3: Evaluate the Second Absolute Value

Evaluate the absolute value of -9. The absolute value of -9 is 9 because the absolute value operator makes all numbers non-negative.

Step 4: Apply the Negative Sign

Apply the negative sign to the absolute value of -9. This gives \(|-9| = 9\), thus \(-|-9| = -9\).

Step 5: Compare the Values

We need to compare -8 and -9. Since -8 is greater than -9 (as -8 is closer to zero), the statement \(-|8| > -9\) is true.

Key Concepts

absolute value

Absolute value is a concept in mathematics that measures the distance of a number from zero on the number line. It is always a non-negative value. For instance, both 8 and -8 have an absolute value of 8, written as \(|8| = 8\) and \(|-8| = 8\).

To determine whether a number is positive or negative, consider its absolute value. The absolute value operation removes any negative sign. This is fundamental in our exercise, as we first convert -8 and 9 to their absolute values before further comparison.

negative numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). For example, -1, -2, and -3 are all negative numbers.

In the context of the given exercise, we compare negative numbers after evaluating absolute values. When comparing negative numbers, the number closer to zero is considered greater. For instance, -8 is greater than -9 because -8 is closer to zero.

This understanding is crucial when we apply the negative sign to our absolute values and perform the final comparison.

inequalities

Inequalities are mathematical expressions that show the relationship between values. They often use symbols like 'greater than' (>) or 'less than' (<).

In our given exercise, we need to compare \(-|8|\textgreater - 9\). This means we need to determine whether the absolute value of \8\ after being made negative is greater than \-9\. Here, our problem becomes comparing negative numbers, where knowing the relationship of distances to zero is essential.

algebraic expressions

Algebraic expressions include variables, numbers, and arithmetic operations.

In the example exercise, \(-|8|\textgreater - 9\), the expression is evaluated step by step. We apply the absolute value operation first, then the negative sign, and finally compare the results. Calculating each part correctly helps in understanding the complete expression and its final comparison.