Problem 39 - Prealgebra

Question

Evaluate each expression if \(x=-12, y=4,\) and \(z=-1\) $$|x|-7$$

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Substitute the values

First, replace the variable \(x\) in the expression \(|x|-7\) with its given value, \(-12\). This gives us the expression: \(|-12|-7\).

Step 2: Calculate the absolute value

Next, calculate the absolute value of \(-12\). The absolute value of a number is its distance from zero on the number line, ignoring the sign. Thus, \(|-12| = 12\).

Step 3: Complete the subtraction

Now, substitute the absolute value back into the expression: \(12 - 7\). Finally, perform the subtraction to get the result: \(12 - 7 = 5\).

Key Concepts

Substitution in Expressions

When tackling mathematical expressions, substitution is a vital skill. It involves replacing variables in an equation with specific values. This allows you to simplify and solve the expression. In this exercise, the variable \(x\) was given the value of \(-12\). By substituting \(x\) with \(-12\) in the expression \(|x| - 7\), we transform it into \(|-12|-7\). This significantly simplifies the problem, making it easier to compute.

• Substitution requires knowing the given values for variables in an expression.
• Substitute each variable with its given number to simplify the process.

Practice using substitution in various expressions to become more comfortable with this foundational step in solving equations.

Properties of Numbers

Understanding the properties of numbers is crucial to evaluate expressions accurately. A core property involved here is the absolute value, which is always a non-negative number.The absolute value of a number is defined as its magnitude, irrespective of direction on the number line. Therefore, \(|-12|\) equals \(12\) because absolute value ignores the negative sign. Knowing that an absolute value represents just how far a number is from zero aids in simplifying expressions involving negative numbers.

• Absolute values result in non-negative outcomes.
• Recognizing properties such as these simplifies the evaluation of expressions.

Mastering number properties can streamline problem-solving by highlighting key simplifications.

Number Line Distance

The concept of distance on a number line plays a vital role in understanding absolute value. Think of absolute value as measuring how far a number is from zero, regardless of direction. This distance is always positive or zero.By considering \(-12\) on a number line, it's located 12 units away from zero, moving left. Thus, the absolute value \(|-12|\) equals 12, demonstrating its distance from zero.

• Absolute value reflects the number's position relative to zero.
• Distances on the number line do not take direction into account, focusing only on space between points.

Understanding the number line and its distances is essential for effectively approaching absolute value problems.