Problem 88 - Intermediate Algebra

Question

In the following exercises, simplify each expression. (a) \(46-(-37)\) (b) \(46+37\)

Step-by-Step Solution

Answer

(a) 83, (b) 83

Step 1: Simplify part (a)

Start by simplifying the expression by addressing the double negative. The expression is: \(46 - (-37)\)Subtracting a negative number is the same as adding the positive of that number: \(46 - (-37) = 46 + 37\).Now, add 46 and 37 together: \(46 + 37 = 83\).

Step 2: Simplify part (b)

Now simplify the expression:\(46 + 37\)Add the two numbers together: \(46 + 37 = 83\).

Key Concepts

Double Negative

Understanding the concept of double negatives is essential in algebra. When you see a double negative in an expression, like \(46 - (-37)\), it's important to remember that two negatives make a positive. This is because subtracting a negative is the same as adding its positive counterpart.
Think of it this way: if owing someone money (negative) is subtracted from what you owe, it's like they are giving you that money, which increases what you have (positive).
For example, \(46 - (-37)\) can be changed to \(46 + 37\), simplifying the problem and making it easier to solve.

Addition of Integers

Addition of integers is a fundamental skill in algebra. When combining two positive integers, you simply add their values.
For instance, in the exercise, we first encountered \(46 + 37\) when converting \(46 - (-37)\), and again directly in part (b) with \(46 + 37\).
Adding integers is straight-forward: you line up the numbers and combine them to get the total.
In our example, we add 46 and 37 to get 83: \[ 46 + 37 = 83 \].
If the integers had different signs, you would subtract the smaller number from the larger number and keep the sign of the larger number.

• Example: \(30 + (-10) = 20\).
• Example: \( -15 + 25 = 10\).

Step-by-Step Solution

Breaking down problems into step-by-step solutions helps to clarify the process and avoids mistakes. Here's another look at simplifying the expression \(46 - (-37)\):
Step 1: Identify the double negative.
Recognize that \(-(-37))\) converts to \(+37\).
Step 2: Rewrite the expression.
Change \(46 - (-37)\) to \(46 + 37\).
Step 3: Add the integers.
Sum up 46 and 37: \(46 + 37 = 83\).
By solving problems step-by-step, you can easily spot where mistakes might occur and ensure each part of the problem is addressed correctly.