Problem 57 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -21+(-17)-11+15-(-10) $$

Step-by-Step Solution

Answer

The simplified expression is -24.

Step 1: Start with Addition of Negative Numbers

The expression \[-21 + (-17) - 11 + 15 - (-10)\]starts with a series of terms that include negative numbers. Begin by combining the first two terms. When adding negative numbers, add their absolute values and keep the negative sign: \[-21 + (-17) = -38\].So the expression becomes: \[-38 - 11 + 15 - (-10)\].

Step 2: Subtract Next Negative Number

Now continue to simplify the next part by subtracting 11 from the already simplified part:\[-38 - 11 = -49\].This makes the expression:\[-49 + 15 - (-10)\].

Step 3: Add Positive Number

Add the positive number 15 to the result from the previous step. When adding, you simply add the numbers directly:\[-49 + 15 = -34\].This simplifies the expression to:\[-34 - (-10)\].

Step 4: Subtracting a Negative Number

Subtracting a negative number is the same as adding its positive counterpart. Therefore:\[-34 - (-10) = -34 + 10\].Now perform the addition:\[-34 + 10 = -24\].

Key Concepts

Addition of Negative Numbers

When adding negative numbers, it's like adding a debt. You are essentially increasing the negative value. To add negative numbers, consider their absolute values, which are the positive versions of the numbers. For instance, if you need to find \(-21 + (-17)\), add the absolute values of 21 and 17, which gives you 38, and keep the negative sign resulting in \(-38\).
Remember:

• Negative + Negative = Larger Negative
• Always add the absolute values first
• Retain the negative sign after addition

Understanding this concept helps us to simplify expressions that contain multiple negative terms by combining them step by step.

Subtracting Negative Numbers

Subtracting a negative number can be tricky at first. It might seem confusing, but the key is to think of it as adding a positive number instead. If you see an expression like \(-34 - (-10)\), interpret the subtraction of a negative as the addition of its absolute value. So, \(-34 - (-10)\) becomes \(-34 + 10\).
This transformation works because two negatives cancel each other out, turning the operation into addition.
Key points to remember:

• "Minus a Negative" is equivalent to "Plus a Positive"
• Focus on flipping the negative sign to a positive when simplifying

Once you understand this, simplifying such expressions becomes straightforward, making it easier to solve complex problems.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions. It involves grouping together terms that share the same variable or, in the case of numbers, just combining plain numbers. In our exercise, you see examples of this mainly when combining the constant terms.
For instance, when you see \(-49 + 15\), you are combining the like terms, which are both just numbers without variables. You perform basic arithmetic to simplify the expression to \(-34\).
Important tips:

• Identify and group terms with common characteristics
• Simplify these groups step by step
• Pay attention to signs when combining (positive with positive, negative with negative)

Practice makes perfect with this concept, helping you to quickly simplify any expression by combining and reducing like terms effectively.