Problem 43 - Algebra 2

Question

Evaluate each expression if \(x=6, y=2.8,\) and \(z=-5\). \(9-|-2 x+8|\)

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Substitute the Values

First, we need to substitute the given values for the variables in the expression. The expression given is \(9 - |-2x + 8|\), and you're told that \(x = 6\). Substitute \(x = 6\) into the expression: \(9 - |-2(6) + 8|\).

Step 2: Simplify Inside the Absolute Value

Next, calculate the expression inside the absolute value. This means calculating \(-2(6) + 8\). First, multiply \(-2\) by \(6\) to get \(-12\). Then add \(8\): \(-12 + 8 = -4\). So now we have \(9 - |-4|\).

Step 3: Evaluate the Absolute Value

Now we simplify the expression by solving the absolute value. The absolute value of \(-4\) is \(4\). Therefore, the expression simplifies to \(9 - 4\).

Step 4: Final Calculation

Finally, perform the subtraction: \(9 - 4 = 5\). This gives us the value of the expression when \(x = 6, y = 2.8,\) and \(z = -5\) is substituted and calculated.

Key Concepts

Substitution Method

The substitution method involves replacing the variables in an expression with their given numeric values. In mathematical exercises, such as evaluating expressions, this step is essential to transition from a generalized formula to a numerical result. The exercise given involves substituting distinct values for the variables involved.

• Identify the variable within the expression. In this case, the variable is \(x\).
• Substitute the numerical value given. For example, if \(x = 6\), then replace every occurrence of \(x\) in the expression with 6.
• Check your substitutions thoroughly to ensure there are no mistakes.

In the provided exercise, substituting the value of \(x\) correctly allows us to progress to further simplifications. This is a necessary basic step for breaking down complex algebraic expressions into simpler arithmetic calculations.

Absolute Value

Absolute value is a key mathematical concept that deals with the magnitude of a number regardless of its sign. It is very intuitive once you understand that it simply measures the distance from zero on the number line, ignoring direction.

• The absolute value of a positive number \(a\) is \(a\) itself.
• The absolute value of a negative number \(-a\) is still \(a\).
• Symbolically, it is represented as \(|a|\), which always results in a non-negative result.

For example, during the given problem, we evaluate \(|-4|\). Since \(-4\) is 4 units away from zero, its absolute value is 4.

Understanding this concept ensures that the final arithmetic on the expression gives an accurate solution. Without knowing absolute values, calculations can lead to incorrect results, especially in expressions involving negative numbers.

Arithmetic Operations

Arithmetic operations are the fundamental building blocks of evaluating any mathematical expression. They include addition, subtraction, multiplication, and division.

• Follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (also from left to right).
• Always simplify expressions inside parentheses or absolute value bars first.

In the given solution, we had to perform multiplication, addition, and subtraction. Multiplication was used to solve \(-2 \times 6\). Then, adding 8 completed the inside of the absolute value. Finally, subtraction was used to obtain the final result: \(9 - 4 = 5\).Each arithmetic step transitions you closer to the final answer. Mastering these operations is essential for efficiently and correctly solving algebraic expressions.