Problem 59 - Elementary Algebra

Question

Simplify. $$ -(-|2|+|-10|) $$

Step-by-Step Solution

Answer

-12

Step 1: Evaluate Absolute Values

The expression has two absolute values: 1. Find \(|2|\), which is the absolute value of 2. Since 2 is positive, \(|2| = 2\).2. Find \(|-10|\), which is the absolute value of -10. This means \(|-10| = 10\).

Step 2: Substitute Absolute Values Back into Expression

Replace the absolute values in the expression: \(-(-|2| + |-10|)\) becomes \(-(-(2) + (10))\).

Step 3: Simplify Inside the Parentheses

The expression now is \(-(2 + 10)\). Calculate \(2 + 10\), which equals 12. So the expression is \(-(12)\).

Step 4: Simplify the Expression

Now, simplify \(-(12)\). The negative sign outside of the parentheses changes the sign of 12, thus becoming -12.

Key Concepts

Understanding Absolute Value

The absolute value of a number is the distance from zero on the number line, regardless of direction. This means that absolute values are always non-negative.
When you see the symbol \(|x|\), it signifies the absolute value of \(x\), which is always positive or zero.
For example, \(|2| = 2\) since 2 is already positive, and \(|-10| = 10\) because the distance from -10 to zero is 10 steps.

• Absolute values help you manage distances and magnitudes without worrying about direction.
• Useful in solving equations where negative numbers might complicate matters.

In your exercise, converting \(-10\) and \(2\) into their absolute values simplifies calculations by removing negative signs.

The Role of Negative Signs

Negative signs in expressions change the direction or sign of a number. When you see a negative sign outside parentheses, it affects everything within.
For example, the expression \[-(x + y)\] means you need to take the negative of the entire sum inside, or \[-x - y\].
In the given problem, you have \[-(-(2) + (10))\]. Although the numbers inside the parentheses, like \(2\) and \(10\), are positive due to absolute values, the outer negative sign changes their sign in the final result.

• Applying multiple negatives can switch signs multiple times.
• Be mindful of parentheses that affect order and application of negative signs.

This method allows you to handle and simplify expressions more effectively.

Simplifying Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a value. Simplification involves reducing an expression to its simplest form.
Typically, you simplify expressions by handling operations systematically:

• Evaluate absolute values.
• Simplify terms inside parentheses.
• Apply negative signs as needed.

In the example, first, replace absolute values and simplify \(2 + 10\) inside parentheses, which gives 12.
Then, apply the negative sign: \(-(12) = -12\).
By following these steps, expressions become easier to work with, helping solve equations and understand mathematical concepts more clearly.