Problem 64 - Intermediate Algebra

Question

In the following exercises, simplify. $$ |17-8|-|13-4| $$

Step-by-Step Solution

Answer

The answer is 0.

Step 1: Evaluate the expressions inside the absolute values

First, calculate the expressions inside the absolute values: \(|17 - 8|\) and \(|13 - 4|\).

Step 2: Calculate the differences

Subtract the numbers inside the absolute values:\(17 - 8 = 9\) and \(13 - 4 = 9\).

Step 3: Apply the absolute value

The absolute value of both results is: \(|9| = 9\).

Step 4: Subtract the absolute values

Finally, subtract the absolute values from each other:\(9 - 9 = 0\).

Key Concepts

simplifying expressions

When simplifying expressions, the goal is to make them as simple as possible while keeping their value the same. Here’s how you can approach this problem step by step:

First, you need to break down the expression into smaller parts so it's easier to handle. For example, let’s look at the problem \(|17-8| - |13-4|\).

Follow these steps:

• Evaluate the expressions inside the absolute values first.
• Solve the subtraction inside those absolute values.
• Then, handle the absolute values themselves.
• Finally, focus on any remaining operations outside the absolute values.

Working step-by-step ensures you don't miss anything and makes the process less confusing.

absolute value

Absolute value is a way to describe how far a number is from zero on the number line, regardless of direction. In mathematical terms, the absolute value of a number \(a\) is denoted as \(|a|\).
If you have a positive number, such as 5, its absolute value is still 5. And if you have a negative number, such as -7, its absolute value is +7.

In our exercise, we deal with expressions \(|17-8|\) and \(|13-4|\).

• The expression \(17-8\) equals 9.
• The expression \(13-4\) equals 9 too.

Therefore, the absolute value of both results is \(|9|\), which is simply 9. Absolute values turn negative results into positive ones and keep positive results the same.

basic subtraction

Subtraction is one of the simplest operations in math. It tells us how much one number is less than another. It can be easily done by removing the second number from the first.

Let’s look at the steps for subtraction with our given example \(17-8\) and \(13-4\).

• First, you subtract 8 from 17, which gives you 9. This is done by counting backwards or directly subtracting the smaller number from the larger one.
• Then, you subtract 4 from 13, which also gives you 9 using the same method.

Now, if both numbers inside absolute values give the same result, these simplified numbers can be directly used in further calculations. In our example, after the subtraction, we get two 9’s. Additionally, subtracting these absolute values (both 9), yields a result of 0.