Problem 72 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ |9-5(1-8)| $$

Step-by-Step Solution

Answer

The value of the expression is 44.

Step 1: Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses. The inner expression is \(1 - 8\). Subtract 8 from 1 to get \(1 - 8 = -7\).

Step 2: Multiply

Next, multiply the result from parentheses by 5 as indicated in the expression. Calculate \(5 \times (-7) = -35\).

Step 3: Subtract the Product from 9

Now focus on the expression \(9 - (-35)\). Subtracting a negative is equivalent to adding the positive, so this becomes \(9 + 35\).

Step 4: Evaluate the Absolute Value

Compute the sum inside the absolute value: \(9 + 35 = 44\). Find the absolute value of 44. Since 44 is already positive, \(|44| = 44\).

Step 5: Final Answer

After following all the steps, the value of the original expression \(|9-5(1-8)|\) is computed to be 44.

Key Concepts

Understanding Parentheses Simplification

When faced with a complex expression, one of the first steps is often to simplify any operations inside parentheses. Parentheses are used in math to group terms that need to be evaluated first, according to the order of operations (PEMDAS/BODMAS).

• P stands for Parentheses
• E for Exponents
• MD for Multiplication and Division
• AS for Addition and Subtraction

If you can identify any calculations within parentheses, handle those first.
For example, in the expression \(|9-5(1-8)|\), focus on the part inside the parentheses: \(1 - 8 \). Here, subtraction needs to be done before any other operations outside the parentheses. Calculating \(1 - 8\), we get \(-7\). This step is crucial because it affects the rest of the operations. Remember, solving what's inside the parentheses correctly sets the stage for everything else to follow.

Multiplication of Integers

Once you have simplified inside the parentheses, the next critical step is to handle any multiplicative operations. Remember that multiplication of integers can involve both positive and negative numbers.
Multiplying two numbers follows these basic rules:

• If both numbers are positive, the result is positive
• If one number is positive and the other is negative, the result is negative
• If both numbers are negative, the result is positive

Take the expression \(5 imes (-7)\). Here, one integer is positive (5) and the other is negative (-7). Since one is positive and the other is negative, the result will be negative: \(-35\). Knowing the sign convention of multiplication helps to avoid mistakes where you might incorrectly assign the wrong sign to your result.

Subtraction Involving Negatives

Working with subtraction can sometimes be tricky, especially when involving negative numbers. Subtraction is essentially the addition of a negative number.
Let's consider the expression \(9 - (-35)\). This can be rewritten as \(9 + 35\) because subtracting a negative number is the same as adding its positive equivalent.
Here’s a simple rule:

• "Subtracting a negative" changes to "adding a positive."

This is because in mathematical terms, two negatives cancel each other out.
In this example, rewriting \(9 - (-35)\) to \(9 + 35\) allows you to carry on the addition straightforwardly, leading to a result of \(44\). Grasping this concept can greatly simplify your calculations in many algebraic expressions.