Problem 22 - Algebra 2

Question

Evaluate each expression if \(a=-5, b=6,\) and \(c=2.8\). \(-|2 c-a|\)

Step-by-Step Solution

Answer

The expression evaluates to -10.6.

Step 1: Identify the expression

The given expression to evaluate is \(-|2c-a|\).We need to substitute the values of \(a\), \(b\), and \(c\) into this expression.

Step 2: Substitute the variables

Replace \(a\) with \(-5\) and \(c\) with \(2.8\) in the expression:\(-|2 \times 2.8 - (-5)|\).

Step 3: Simplify inside the absolute value

Calculate \(2 \times 2.8 = 5.6\).Now the expression inside the absolute value becomes:\(|5.6 - (-5)|\).Simplify further:\(5.6 - (-5) = 5.6 + 5 = 10.6\).

Step 4: Apply the absolute value

Calculate the absolute value:\(|10.6| = 10.6\).Since the absolute value of a positive number is the number itself, we have\(10.6\) as the result of the absolute value.

Step 5: Apply the negative sign

Now apply the negative sign outside the absolute value to get the final result:\(-10.6\).

Key Concepts

Understanding Absolute Value

Absolute value refers to the distance of a number from zero on a number line. It is always a non-negative number. For any real number, say \( x \), the absolute value is denoted by \( |x| \). This operation ignores whether the original number is positive or negative.
To compute the absolute value, you simply:

• Ignore the sign of the number inside the absolute value bars.
• Always result in a positive number or zero.

For instance, \( |-5| = 5 \) because 5 is five units away from zero on the number line, ignoring the negative sign. In the problem we are solving, when we encounter an expression like \( |10.6| \), it remains \( 10.6 \). The absolute value does not alter numbers that are positive.

The Process of Substitution

Substitution involves replacing variables in an expression with given numerical values. This technique is essential in making abstract algebraic expressions concrete and numerical.
In our exercise, the function involves variables \( a \), \( b \), and \( c \). To find the specific value of the expression \(-|2c-a|\), we substitute:

• \( a \) with \(-5\)
• \( c \) with \(2.8\)

The expression becomes \(-|2 \times 2.8 - (-5)|\). This substitution transforms the variables into numbers, setting the stage for further simplification. Substitution helps in evaluating the expression with the specific values for variables provided.

Simplifying Algebraic Expressions

Simplification is the process of reducing an expression to its simplest form. This step is crucial for clarity and ease of calculation.
In our exercise, once we substitute the values, we simplify inside the absolute value: Multiply \(2 \times 2.8\) to get \(5.6\). Then we simplify the expression further: \(5.6 - (-5)\). This subtraction becomes an addition because subtracting a negative is the same as adding its positive: \(5.6 + 5 = 10.6\).
The goal of simplification is to reduce complexity, making the calculation straightforward. It helps to eliminate unnecessary steps and provides a direct pathway to the final result.

Handling Negative Numbers

Negative numbers are values less than zero, often represented with a minus sign. They can sometimes be tricky, especially when performing operations such as subtraction and multiplication.
When you see a double negative in subtraction, like \(5.6 - (-5)\), it converts to an addition: \(5.6 + 5\). This is a fundamental rule: subtracting a negative number results in adding its positive counterpart.
In the final step of our problem, applying a negative outside \(|10.6|\) changes the sign from positive to negative, resulting in \(-10.6\). Understanding how negative numbers interact within expressions ensures mathematical accuracy.