Problem 2 - Algebra 2

Question

Evaluate each expression if \(a=-4\) and \(b=1.5\). \(|-6 b|\)

Step-by-Step Solution

Answer

The evaluated expression is 9.

Step 1: Substitute Values

First, substitute the given value of \(b\) into the expression. The expression given is \[|-6b|.\] Substituting the value of \(b = 1.5\), we get \[|-6 \times 1.5|.\]

Step 2: Multiply

Next, calculate the multiplication inside the absolute value. So, compute \(-6 \times 1.5\). This gives us \(-9.\)

Step 3: Apply Absolute Value

Now, apply the absolute value operator to the result. The absolute value of a number is its distance from zero on the number line, regardless of direction. Since \(-9\) is the result from the multiplication, we get \[|-9| = 9.\]

Key Concepts

Substitution

Substitution is an essential mathematical technique where we replace a variable with a value. In our exercise, we are given an expression with variables, and the task requires us to calculate its value for certain assignments. This involves replacing each occurrence of the variable with the given value.

Here, the expression is \(|-6b|\) with the variable \(b\) assigned the value of \(1.5\). By substituting \(b = 1.5\) into the expression, we transform it into \[|-6 \times 1.5|\]This adjustment allows us to proceed with evaluating the expression based solely on arithmetic operations. Substitution simplifies expressions to a form that can be easily solved.

Multiplication

After substitution, our task is to engage in multiplication. Multiplication is one of the basic arithmetic operations and involves scaling one number by another.

In our exercise, the expression became \[|-6 \times 1.5|\]Our next step is to multiply \(-6\) by \(1.5\).

• First, think of \(-6\) as moving \(6\) units in the negative direction.
• When we multiply 6 by 1.5, we get \(9\). Since it was \(-6\) originally, the result becomes \(-9\).

Thus, the multiplication step gives us \(-9\).

It is crucial to attend to the signs during multiplication: a negative times a positive results in a negative.

Distance from Zero

The concept of absolute value is deeply tied to an understanding of distance, specifically 'distance from zero' on the number line. The absolute value of any number, positive or negative, reflects how far it is from zero, disregarding its direction.

When we calculated \-6 \times 1.5\ and obtained \(-9\), we now need to evaluate \(|-9|\).

• Ignore the sign of \(-9\) and simply consider its distance to zero.
• The distance from \(-9\) to zero is \(9\) units on the number line.

Thus, the result of \(|-9|\) is \(9\). Absolute values turn negative results into positive, aligning well with the notion of distance which is inherently non-negative.