Problem 45 - Algebra 2

Question

Evaluate each expression if \(x=6, y=2.8,\) and \(z=-5\). \(|z-x|-|10 y-z|\)

Step-by-Step Solution

Answer

The result is -22.

Step 1: Substitute the Values into the Expression

First, substitute the given values of \(x\), \(y\), and \(z\) into the expression. The expression is \[|z-x|-|10y-z|\].Substitute \(x = 6\), \(y = 2.8\), and \(z = -5\):- \(|z-x| = |-5 - 6|\)- \(|10y-z| = |10 \times 2.8 - (-5)|\)

Step 2: Calculate the Absolute Values

The next step is to compute the absolute values for each expression:1. \(|z-x| = |-5 - 6| = |-11| = 11\)2. \(|10y-z| = |28 + 5| = |33| = 33\).

Step 3: Subtract the Absolute Values

Subtract the absolute value obtained from the second term from the first:- Compute \(11 - 33 = -22\).

Key Concepts

Absolute Value

When you see the term **absolute value**, it may sound a bit complex, but it is actually quite straightforward. The absolute value of a number is simply its distance from zero on the number line, and it is always positive. Think of it as stripping away any negative sign a number has, leaving the pure magnitude. This is represented with vertical bars around the number or expression, like this: \(|-11|\).

Absolute value operations always yield non-negative results. For instance, the absolute value of both -5 and 5 is 5. This is because they are both five units away from zero.

In the example given, you calculated the absolute value of expressions like \(|z-x| = |-11|\) which resulted in 11. This step makes sure all numbers are "positive" before any further arithmetic is done.

Substituting Values

In algebraic expressions, **substituting values** is a key step used to solve problems involving variables. It allows for evaluating expressions with specific numbers.

To substitute a value, simply replace the variable in the expression with the number provided. In the given exercise, the expression \(|z-x|-|10y-z|\) needed numbers for \(x\), \(y\), and \(z\). Their respective values were given as 6, 2.8, and -5.

This leads to the calculation:

• For \(|z-x|\), substitute: \(|-5 - 6|\)
• For \(|10y-z|\), this substitution becomes \(|10 \times 2.8 + 5|\)

Substituting values transforms abstract expressions into numerical calculations, facilitating further computation.

Arithmetic Operations

**Arithmetic operations** involve basic math functions such as addition, subtraction, multiplication, and division. In solving the given expression, subtraction played a vital role.

After computing absolute values for the expressions, namely 11 and 33 in this exercise, the next arithmetic step was to subtract these two values from each other. The subtraction principle is applied here: always ensure that the minuend (first number) minus the subtrahend (second number) is handled correctly.

Therefore, you calculated:

• Subtract \(\ 33\ from \ 11\), yielding \(11 - 33 = -22\).

This results in the final answer of the evaluated expression. Arithmetic operations help manipulate numbers to achieve a solution, showcasing the beauty of algebraic problem-solving.