Problem 19 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$|-2| x+|-8| x+|10| x$$

Step-by-Step Solution

Answer

The simplified expression is \(20x\).

Step 1: Simplify Absolute Values

First, simplify the absolute values in the expression. Since the absolute value function converts any number into its non-negative form, we have \(-2\) becomes \(2\), \(-8\) becomes \(8\), and \(10\) remains \(10\). So the expression becomes \(2x + 8x + 10x\).

Step 2: Combine Like Terms

Now, combine the coefficients of the like terms \(x\). Add the coefficients: \(2x + 8x + 10x = (2 + 8 + 10)x = 20x\).

Step 3: Write the Simplified Expression

The simplified expression is \(20x\). This means that the original expression simplifies to ten times \(x\).

Key Concepts

Absolute Value

The concept of absolute value may initially seem complex, but it is quite simple once you get the hang of it. Absolute value refers to the distance of a number from zero on a number line, and it always results in a positive number. Imagine you have

• -2
• -8
• 10

in your expression. For these numbers:

• The absolute value of default is 2 because its distance from zero is 2 units.
• Similarly, absolute value of -8 is 8.
• 10 is already positive, so its absolute value remains 10.

The critical step in simplifying expressions is first converting all negative numbers into their absolute, positive equivalents. So, whenever you spot the absolute value operator, always think about the direction on the number line and distance from zero.

Like Terms

Like terms in algebra are terms that have identical variables and exponents. They allow us to simplify expressions easily. Let’s look at the expression given: ul> li> 2x li> 8x li>10x All of these terms are like terms because they all have the variable 'x.' This is crucial because to combine terms, they need to be 'like' or similar to one another.
Like terms make it easier to simplify because you only need to add or subtract the coefficients, which are the numbers in front of the variables. In this case: 2, 8, and 10.

Simplifying Expressions

Simplifying expressions is a key part of algebra, making complex expressions easier to work with and understand. After converting all terms to absolute values and identifying like terms, the next step is simplifying.
To simplify, you add up all the coefficients of the like terms:

• Start with 2x and add 8x, resulting in 10x.
• Then, add 10x, bringing the sum to 20x.

This combination of like terms results in a single, simpler expression:
20x. By understanding how to simplify, you'll reduce the chance of errors in algebraic calculations and see solutions faster. Plus, it's a vital skill for everything from solving equations to calculus.