Problem 9 - Elementary Algebra

Question

Simplify the following problems. $$ -(-|12|) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression $-(-|12|)$ is 12.

Step 1: Compute the Absolute Value

First, calculate the absolute value of the given number. To do this, find the distance of the number from zero on a number line, disregarding its sign. In our case, we are given the absolute value of 12: $$ |12| = 12 $$

Step 2: Negate the Result

Now, apply the negative sign outside of the absolute value. As the absolute value of a number is always non-negative, applying the negative sign will negate the result. So in our case: $$ -|12| = -12 $$

Step 3: Simplify Double Negatives

Finally, we need to simplify the double negative in the expression. A double negative means multiplying two negative signs, which turns the result into a positive value. So, our final simplification is: $$ -(-12) = 12 $$ Thus, the simplified expression is 12.

Key Concepts

Understanding Negation in Algebra

Negation in algebra involves changing the sign of a number or expression. If you have a positive number and you apply negation, it becomes negative, and vice versa. Essentially, negation is like flipping a switch that changes the sign of a numerical value. Consider it as multiplying the number by -1.

For example:

• If you have the number 5, the negation of 5 is -5, which is calculated as \(5 \times (-1)\).
• Similarly, if you start with -7, its negation would be \(-7 \times (-1) = 7\).

Negation is an essential concept because it helps to solve equations and understand the behavior of different algebraic expressions when applied to specific operations like subtraction and division. It can significantly affect the result of calculations, making this foundational understanding crucial for advancing in algebra.

The Concept of Double Negatives

Double negatives in algebra occur when you have two negative signs next to each other or one acting on the other, essentially multiplying each other. When you encounter two negatives, they make a positive.

This principle can be confusing for beginners, but it's helpful to remember that multiplying two negatives in mathematics negates each other, leading to a positive outcome. Think of it like two wrongs making a right, a familiar rule in many languages!

• For instance, $-(-7) = 7$. The negative sign outside the parenthesis cancels out the negative inside.
• Similarly, $-(-x)$ simplifies to $x$ for any variable x.

This concept is crucial when simplifying expressions because it can impact the overall sign and outcome of mathematical equations. Simplifying expressions with double negatives is often necessary in solving algebraic problems.

Simplification in Algebra

Simplification in algebra means rewriting expressions or equations in their simplest form. This often involves combining like terms, reducing fractions, eliminating unnecessary symbols, and applying rules like negation and handling double negatives effectively.

The main goal of simplification is to make mathematical expressions easier to understand and work with. It also helps in revealing the underlying structure of expressions which might not be immediately apparent. Here are some key points to consider:

• Always start by solving any expressions inside parentheses.
• Apply negation and handle double negatives early on to avoid confusion.
• Combine like terms by adding or subtracting similar terms to get a coherent result.

For example:

• If you have the expression $3x + 5 - 2x - 5$, the simplified form is $x$ as you combine $3x - 2x$ and $5 - 5$.
• If dealing with fractions, reduce by dividing by common factors. Understand that reduction makes solving equations more straightforward.

Mastering simplification is a core skill in algebra, facilitating more complex operations and problem-solving tasks as algebra becomes more advanced.