Lesson 7: Simplifying and Comparing Expressions with Symbols of Inclusion

New Concept

The journey into algebra begins with a single idea: a variable is used to represent an unknown number. This lets us build and analyze expressions.

What’s next

First, we will master the rules for order. Get ready for worked examples that break down simplifying expressions with parentheses, brackets, and fraction bars.

Property

Symbols of inclusion, such as fraction bars, absolute-value symbols, parentheses, braces, and brackets, indicate that the enclosed parts are a single term. To simplify, begin inside the innermost symbol of inclusion and work outward, always following the order of operations.

Examples

$20 - [10 - (2+5)] = 20 - [10 - 7] = 20 - 3 = 17$
$5 \cdot [3 + (4-2)^2] = 5 \cdot [3 + 2^2] = 5 \cdot [3+4] = 5 \cdot 7 = 35$
$\{16 \div [2 \cdot (8-6)]\} = \{16 \div [2 \cdot 2] \} = \{16 \div 4\} = 4$

Explanation

Think of these symbols as a secret mission! Your first task is always to solve the puzzle hidden deep inside the innermost parentheses. Once you crack that code, you can work your way out to solve the bigger mystery. It’s all about starting from the inside!

Property

To simplify expressions with absolute-value symbols, first perform the operations inside the symbols. Then, take the absolute value of the result. For example, $|a-b|$ requires finding the value of $a-b$ first, then making the result positive.

Examples

$25 - |10 - 15| = 25 - |-5| = 25 - 5 = 20$
$3 \cdot |2 - 9| + 1 = 3 \cdot |-7| + 1 = 3 \cdot 7 + 1 = 21 + 1 = 22$

Explanation

Absolute value bars are like a positivity machine! No matter what happens inside—addition, subtraction, or just a single negative number—the final value that comes out must be positive. It measures distance from zero, which can't be negative!

Property

When simplifying a rational expression like $\frac{6-3}{4-2}$, the fraction bar acts as a symbol of inclusion. The numerator and denominator must be simplified into single numbers first before the final division is performed.

Examples

$\frac{(5+3)^2}{10-2} = \frac{8^2}{8} = \frac{64}{8} = 8$
$[2 \cdot (3+1)^2] - \frac{6 \cdot 5}{3} = [2 \cdot 4^2] - \frac{30}{3} = [2 \cdot 16] - 10 = 32 - 10 = 22$

Explanation

A fraction bar is a secret grouping tool that creates two separate zones: top and bottom. You have to completely solve everything on the top floor and everything on the bottom floor before you can see how they relate through division. It’s an upstairs-downstairs problem!