Problem 41 - Prealgebra

Question

Evaluate each expression if \(x=-12, y=4,\) and \(z=-1\) $$|z|-|x|$$

Step-by-Step Solution

Answer

The value of the expression is -11.

Step 1: Substitute Values into Expression

First, substitute the given values for the variables into the expression. The expression is \(|z| - |x|\). Plug in \(z = -1\) and \(x = -12\). This gives us \(|-1| - |-12|\).

Step 2: Evaluate Absolute Values

Calculate the absolute values: \(|-1| = 1\) and \(|-12| = 12\).

Step 3: Subtraction of Absolute Values

Subtract the absolute values calculated: \(1 - 12 = -11\).

Key Concepts

Absolute Value

The concept of absolute value is important in prealgebra. It helps understand the magnitude of a number, regardless of its sign. In simpler terms, the absolute value of a number is its distance from zero on the number line. Here's how absolute value works:

• For any positive number or zero, the absolute value is the number itself. For example, \( |4| = 4 \).
• For any negative number, the absolute value is the number without the negative sign. For example, \( |-4| = 4 \).

When working with expressions, treat absolute value bars like parentheses. Handle operations inside them first, then take the absolute value of the result. Understanding absolute value is crucial when dealing with any mathematical expressions that involve negative numbers, such as \( |-1| \) or \( |-12| \). These result in positive values, \( 1 \) and \( 12 \) respectively, which significantly change how you evaluate expressions.

Variable Substitution

Variable substitution is another essential concept in math, especially in algebra. It involves replacing variables in an expression or equation with given numbers to simplify and solve problems. When given an exercise with specific variable values, follow these steps:

• Identify all variables in the expression.
• Replace each variable with the given number.
• Ensure all replacements are accurate before proceeding with calculations.

Using variable substitution is like decoding a message. For the expression \(|z| - |x|\), you substitute \(z = -1\) and \(x = -12\) to get \(|-1| - |-12|\). This substitution allows you to work with concrete numbers, making the math much more straightforward. A solid grasp of variable substitution skills can help solve complex problems by reducing unknowns.

Integer Operations

Integer operations refer to the arithmetic procedures used on whole numbers, both positive and negative. In prealgebra, learning integer operations helps solve various types of equations efficiently. Here are the basic operations:

• Addition: When adding integers with the same sign, add their absolute values and keep the common sign. For different signs, subtract the smaller absolute value from the larger one, and take the sign of the number with the larger absolute value.
• Subtraction: To subtract integers, add the opposite of the number being subtracted. For example, \(3 - (-2)\) becomes \(3 + 2\).
• Multiplication and Division: Multiplying or dividing two numbers with the same sign results in a positive number. If the signs differ, the result is negative.

In our example, after calculating the absolute values, you subtract \(12\) from \(1\), resulting in \(-11\). It is important to execute these operations in order to derive the correct answer. Knowing how to handle positive and negative integers can simplify tackling any mathematical expression.