Problem 34 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ -4-(-15) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $$11$$.

Step 1: Remove parentheses

To simplify the expression, begin by removing the parentheses. Recall that subtracting a negative number is the same as adding its opposite. So, $$-4-(-15) = -4 + 15$$

Step 2: Perform the operation

Now, we just need to perform the addition: $$-4 + 15 = 11$$

Step 3: Final Answer

The solution to the expression is $$11$$.

Key Concepts

Subtraction of Negative Numbers

Subtracting negative numbers might initially seem confusing, but it's simpler than it appears. The key is to remember that subtracting a negative is equivalent to adding its positive counterpart.

For example, in the expression \(-4 - (-15)\), the \(-(-15)\) is the same as \(+15\). Thus, \(-4 - (-15)\) becomes \(-4 + 15\). This conversion helps eliminate the double negative, simplifying the operation.

When you encounter two negatives back-to-back, think of changing the double negative to a positive. This is a crucial part of understanding and simplifying expressions involving negative numbers.

This rule helps you tackle complex equations with ease and ensures clarity in problem-solving.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves rewriting an expression in its simplest form. This often means eliminating unnecessary brackets and reducing the number of operations needed.

In the example \(-4 - (-15)\), simplification happens by removing parentheses: \(-4 + 15\). This makes it much easier to see what the next steps are.

• Identify and eliminate double negatives: As demonstrated, transforming \(-(-15)\) into \(+15\) reduces complexity.
• Perform straightforward operations first: Once simplified, proceed with calculations like addition or subtraction.

This process reduces the likelihood of mistakes and misunderstandings, paving the way for accurate and quick solutions.

Addition of Integers

Adding integers is a fundamental math skill that requires understanding how numbers combine on a number line. It involves straightforward rules based on the sign of the integers involved.

When adding positive and negative numbers, visualize a number line:

• If numbers share the same sign (both positive or both negative), add their absolute values and keep the common sign.
• If one is positive and the other is negative, subtract the smaller absolute value from the larger one, keeping the sign of the number with the larger absolute value.

For instance, in \(-4 + 15\), you'll think:
- Start at -4 on the number line.
- Since 15 is positive, move 15 steps to the right.
- Land at 11, which is your answer.

Understanding and applying these rules can help simplify increasingly complex expressions, making problem-solving more intuitive and less reliant on rote memorization.