Problem 21 - Algebra 2

Question

Evaluate each expression if \(a=-5, b=6,\) and \(c=2.8\). \(-|18-5 c|\)

Step-by-Step Solution

Answer

The evaluated expression is -4.

Step 1: Substitute the Given Values

First, substitute the given values into the expression. We have the expression \[-|18 - 5c|\]. Replacing \(c\) with 2.8, the expression becomes \[-|18 - 5(2.8)|\].

Step 2: Simplify Inside the Absolute Value

Simplify the expression inside the absolute value. This involves calculating \[18 - 5 imes 2.8\]. First, compute \[5 \times 2.8 = 14\]. Then subtract: \[18 - 14 = 4\].

Step 3: Apply the Absolute Value

Use the absolute value operation: \[|4| = 4\]. Since the number inside the absolute value was positive, the absolute value of 4 is simply 4.

Step 4: Apply the Negative Sign

Finally, include the negative sign in front of the absolute value. This gives \[-(4) = -4\].

Key Concepts

Understanding Substitution in Algebra

Substitution is a powerful tool in algebra, which allows us to replace variables in an expression with given numerical values. This simplifies the problem and makes it easier to solve.
Let's break it down:

• Identify the variables present in the expression. In our case, we have the variable \(c\).
• Whenever the variable \(c\) appears, we substitute \(c = 2.8\).

By doing this, we transform the expression from an unknown form to a specific calculation. This step is crucial—as it takes an abstract expression and turns it into concrete numbers that we can then easily manipulate. Without substitution, we would be unable to progress towards a solution.

Simplifying Algebraic Expressions

Once we substitute the given values, it's time to simplify the expression. Simplification makes complex expressions more manageable, enabling us to understand and solve them. To simplify:

• Perform any multiplications or divisions first. In our problem, we calculated \(5 \times 2.8\) to get \(14\).
• Next, handle any additions or subtractions. Subtract \(14\) from \(18\) to get \(4\).

The key to simplification is performing calculations step-by-step, following the correct order of operations (PEMDAS/BODMAS). This approach ensures you deal with elements collectively, leading to a cleaner, more simplified expression.

Working with Negative Numbers

Negative numbers can sometimes add a layer of complexity to mathematical problems. It's important to understand how negative signs affect expressions. In our example:

• After obtaining the simplified expression within the absolute value, we noted it as \(|4|\).
• Applying the absolute value results in \(4\), since the absolute value of a positive number stays the same.
• The absolute value symbol is removed by including the negative sign from the original expression, resulting in \(-4\).

This demonstrates how negative signs are handled after determining whether numbers inside absolute values are positive or negative.

Algebraic Expressions and Absolute Values

Algebraic expressions are combinations of numbers, variables, and operations. They're like sentences in the language of mathematics. When they include absolute values, they ask us to consider only the positive magnitude of a number. Here’s how it works:

• The absolute value \(|x|\) of a number \(x\) is its distance from zero on the number line. It's always non-negative.
• When evaluating absolute values, ignore any original negative signs before the absolute value operation.
• Once the expression inside the absolute value is evaluated, then include any external negative signs, like we did converting \(|4|\) into \(-4\).

Absolute values are key in ensuring expressions accurately reflect real-world constraints, omitting negative magnitudes that physically can't exist.