1. Write Variable Expressions: One Operation

Definition: A variable expression uses numbers, variables (letters), and operation symbols to represent a mathematical relationship.

Key Words and Their Operations:

Translation Examples:

• The sum of \(x\) and 7: \(x + 7\)
• 9 less than \(a\): \(a - 9\)
• The product of 5 and \(b\): \(5b\)
• \(c\) divided by 2: \(\frac{c}{2}\)
• Twice \(d\): \(2d\)
• Half of \(e\): \(\frac{e}{2}\) or \(\frac{1}{2}e\)

Important Note: When writing "less than" or "subtracted from," reverse the order!

"5 less than \(x\)" means \(x - 5\) (NOT \(5 - x\))

2. Write Variable Expressions: Two or Three Operations

Strategy: Break down complex phrases into parts and identify each operation separately. Use parentheses when necessary.

Two Operations Examples:

• 5 times a number, increased by 3: \(5x + 3\)
• 7 more than twice a number: \(2n + 7\)
• The difference between 10 and 3 times \(y\): \(10 - 3y\)
• Half of a number minus 6: \(\frac{x}{2} - 6\)
• The product of 4 and \(a\), divided by 5: \(\frac{4a}{5}\)

Expressions with Parentheses:

Use parentheses when the order of operations matters!

• 3 times the sum of \(x\) and 5: \(3(x + 5)\)
• Twice the difference of \(n\) and 8: \(2(n - 8)\)
• The sum of \(a\) and \(b\), divided by 3: \(\frac{a + b}{3}\)

Three Operations Examples:

• 5 times the sum of \(x\) and 3, minus 7: \(5(x + 3) - 7\)
• 3 more than twice the difference of \(y\) and 4: \(2(y - 4) + 3\)
• The quotient of 8 and \(n\), plus 12: \(\frac{8}{n} + 12\)

Key Tip: Look for commas in word problems—they often indicate where to use parentheses or separate operations.

3. Write Variable Expressions from Diagrams

Strategy: Analyze visual representations (geometric shapes, tape diagrams, bar models) to write algebraic expressions.

Common Diagram Types:

1. Tape Diagrams (Bar Models):

• Count the number of equal parts (variable sections)
• Note any additional constant values added or subtracted
• Example: Three boxes labeled \(x\) plus a box with 5 → \(3x + 5\)

2. Geometric Diagrams:

• Perimeter: Add all sides
• Rectangle perimeter: If length = \(l\) and width = \(w\), then \(P = 2l + 2w\)
• Area of rectangle: \(A = l \times w\) or \(lw\)

3. Number Line Diagrams:

• Identify the starting point and endpoint
• Calculate the distance or difference
• Example: From \(x\) to \(x + 7\) represents an increase of 7

Example Problems:

Diagram: A rectangle with length \(x + 4\) and width \(x\)

Perimeter: \(2(x + 4) + 2x = 2x + 8 + 2x = 4x + 8\)

Area: \(x(x + 4) = x^2 + 4x\)

4. Write Variable Expressions: Word Problems

Steps to Solve:

1. Read carefully and identify what the variable represents
2. Identify key words that indicate operations
3. Determine the order of operations
4. Write the expression using proper notation
5. Check if your expression makes sense in context

Real-World Examples:

Example 1: Sarah has 5 more apples than Tom. If Tom has \(t\) apples, write an expression for Sarah's apples.

Answer: \(t + 5\)

Example 2: A taxi charges $3 for the first mile and $2 for each additional mile. If you travel \(m\) additional miles, write an expression for the total cost.

Answer: \(3 + 2m\)

Example 3: The length of a rectangle is 3 more than twice its width. If the width is \(w\), write an expression for the length.

Answer: \(2w + 3\)

Example 4: A store offers a discount of $10 on any purchase over $50. If the original price is \(p\) dollars (where \(p > 50\)), write an expression for the sale price.

Answer: \(p - 10\)

Example 5: Concert tickets cost $25 each. If you buy \(n\) tickets and pay a $5 processing fee, write an expression for the total cost.

Answer: \(25n + 5\)

5. Evaluate One-Variable Expressions

Definition: To evaluate means to find the numerical value of an expression by substituting a given value for the variable.

Steps to Evaluate:

1. Substitute the given value for the variable (use parentheses!)
2. Follow order of operations (PEMDAS/BODMAS)
3. Simplify to get a single number

Order of Operations (PEMDAS):

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Examples:

Example 1: Evaluate \(3x + 7\) when \(x = 4\)

\(3(4) + 7 = 12 + 7 = 19\)

Example 2: Evaluate \(2y^2 - 5y + 3\) when \(y = 3\)

\(2(3)^2 - 5(3) + 3\)

\(= 2(9) - 15 + 3\)

\(= 18 - 15 + 3 = 6\)

Example 3: Evaluate \(\frac{n + 8}{4}\) when \(n = 12\)

\(\frac{12 + 8}{4} = \frac{20}{4} = 5\)

Important: Always use parentheses when substituting negative numbers or expressions!

6. Evaluate Multi-Variable Expressions

Strategy: Substitute values for ALL variables, then simplify using order of operations.

Steps:

1. Replace each variable with its given value (in parentheses)
2. Simplify using PEMDAS
3. Calculate the final answer

Examples:

Example 1: Evaluate \(2a + 3b\) when \(a = 5\) and \(b = 4\)

\(2(5) + 3(4) = 10 + 12 = 22\)

Example 2: Evaluate \(x^2 - 2xy + y^2\) when \(x = 3\) and \(y = 2\)

\((3)^2 - 2(3)(2) + (2)^2\)

\(= 9 - 12 + 4 = 1\)

Example 3: Evaluate \(\frac{a + b}{c}\) when \(a = 8\), \(b = 10\), and \(c = 3\)

\(\frac{8 + 10}{3} = \frac{18}{3} = 6\)

Example 4: Evaluate \(3m^2 + 2mn - n^2\) when \(m = 4\) and \(n = -2\)

\(3(4)^2 + 2(4)(-2) - (-2)^2\)

\(= 3(16) + 2(4)(-2) - 4\)

\(= 48 - 16 - 4 = 28\)

Common Mistake: Forgetting to use parentheses with negative numbers can lead to sign errors!

7. Evaluate Absolute Value Expressions

Definition: The absolute value of a number is its distance from zero on the number line. It's always non-negative.

Absolute Value Notation:

\(|x|\) = absolute value of \(x\)

• If \(x \geq 0\), then \(|x| = x\)
• If \(x < 0\), then \(|x| = -x\) (which makes it positive)

Examples of absolute values:

• \(|5| = 5\)
• \(|-5| = 5\)
• \(|0| = 0\)
• \(|-12| = 12\)

Steps to Evaluate:

1. Substitute the value for the variable
2. Simplify inside the absolute value bars first
3. Take the absolute value (make it positive)
4. Complete any remaining operations

Examples:

Example 1: Evaluate \(|x - 5|\) when \(x = 2\)

\(|2 - 5| = |-3| = 3\)

Example 2: Evaluate \(3|y| + 7\) when \(y = -4\)

\(3|-4| + 7 = 3(4) + 7 = 12 + 7 = 19\)

Example 3: Evaluate \(|2n + 1|\) when \(n = -3\)

\(|2(-3) + 1| = |-6 + 1| = |-5| = 5\)

Example 4: Evaluate \(|a| - |b|\) when \(a = -8\) and \(b = -5\)

\(|-8| - |-5| = 8 - 5 = 3\)

Key Point: Absolute value bars act like parentheses—simplify inside them first!

8. Evaluate Radical Expressions

Definition: A radical expression contains a root symbol. The most common is the square root (\(\sqrt{}\)).

Radical Notation:

• \(\sqrt{x}\) = square root of \(x\) (what number squared equals \(x\)?)
• \(\sqrt[3]{x}\) = cube root of \(x\) (what number cubed equals \(x\)?)
• \(\sqrt[n]{x}\) = \(n\)th root of \(x\)

Common Square Roots (Memorize These!):

Steps to Evaluate:

1. Substitute the value for the variable
2. Simplify inside the radical first
3. Find the root
4. Complete any remaining operations

Examples:

Example 1: Evaluate \(\sqrt{x}\) when \(x = 49\)

\(\sqrt{49} = 7\)

Example 2: Evaluate \(\sqrt{2n + 7}\) when \(n = 9\)

\(\sqrt{2(9) + 7} = \sqrt{18 + 7} = \sqrt{25} = 5\)

Example 3: Evaluate \(3\sqrt{a} - 5\) when \(a = 16\)

\(3\sqrt{16} - 5 = 3(4) - 5 = 12 - 5 = 7\)

Example 4: Evaluate \(\sqrt{b^2 + 9}\) when \(b = 4\)

\(\sqrt{(4)^2 + 9} = \sqrt{16 + 9} = \sqrt{25} = 5\)

Note: The square root symbol acts like parentheses—simplify what's underneath first!

9. Evaluate Rational Expressions

Definition: A rational expression is a fraction where the numerator and/or denominator contains a variable.

General Form:

\(\frac{P(x)}{Q(x)}\) where \(Q(x) \neq 0\)

Steps to Evaluate:

1. Substitute the value for the variable in both numerator and denominator
2. Simplify the numerator
3. Simplify the denominator
4. Divide (simplify the fraction if possible)

Examples:

Example 1: Evaluate \(\frac{x + 3}{x - 2}\) when \(x = 5\)

\(\frac{5 + 3}{5 - 2} = \frac{8}{3}\)

Example 2: Evaluate \(\frac{2n}{n + 4}\) when \(n = 6\)

\(\frac{2(6)}{6 + 4} = \frac{12}{10} = \frac{6}{5}\)

Example 3: Evaluate \(\frac{y^2 - 9}{y + 3}\) when \(y = 5\)

\(\frac{(5)^2 - 9}{5 + 3} = \frac{25 - 9}{8} = \frac{16}{8} = 2\)

Example 4: Evaluate \(\frac{3a - 7}{2a + 1}\) when \(a = 4\)

\(\frac{3(4) - 7}{2(4) + 1} = \frac{12 - 7}{8 + 1} = \frac{5}{9}\)

Important Restriction: The denominator can NEVER equal zero. Always check that your substitution doesn't make the denominator 0!

Example: \(\frac{x + 2}{x - 5}\) is undefined when \(x = 5\) because the denominator would be zero.

10. Identify Terms and Coefficients

Key Definitions:

Term: A single number, variable, or product of numbers and variables separated by + or − signs.

Example: In \(3x^2 + 5x - 7\), the terms are \(3x^2\), \(5x\), and \(-7\)

Coefficient: The numerical factor of a term containing a variable.

Example: In \(3x^2\), the coefficient is 3

Constant: A term without a variable.

Example: In \(3x^2 + 5x - 7\), the constant is \(-7\)

Variable: A letter representing an unknown quantity.

Example: In \(3x^2 + 5x - 7\), the variable is \(x\)

Like Terms: Terms that have the same variable(s) raised to the same power(s).

Example: \(3x\) and \(5x\) are like terms; \(3x^2\) and \(5x\) are NOT like terms

Detailed Examples:

Expression: \(7a^2 - 3a + 9\)

• Terms: \(7a^2\), \(-3a\), \(9\)
• Coefficients: 7 (for \(a^2\)), -3 (for \(a\))
• Constant: 9
• Variable: \(a\)

Expression: \(4xy + 2x - 5y + 6\)

• Terms: \(4xy\), \(2x\), \(-5y\), \(6\)
• Coefficients: 4 (for \(xy\)), 2 (for \(x\)), -5 (for \(y\))
• Constant: 6
• Variables: \(x\) and \(y\)

Expression: \(-n + 8\)

• Terms: \(-n\), \(8\)
• Coefficient: -1 (for \(n\)) — When no number is shown, the coefficient is 1 or -1
• Constant: 8

Special Cases:

• \(x\): The coefficient is 1 (understood)
• \(-x\): The coefficient is -1
• \(\frac{x}{2}\): The coefficient is \(\frac{1}{2}\)

11. Sort Factors of Variable Expressions

Definition: A factor is a number or expression that is multiplied to form a product.

Understanding Factors:

In \(5xy\), the factors are: 5, \(x\), and \(y\)

We can also write: \(5xy = 5 \cdot x \cdot y\)

Types of Factors:

• Numerical Factor: The number part (coefficient)
• Variable Factor: The variable part(s)
• Literal Factor: Another term for variable factors

Examples of Identifying Factors:

Example 1: \(12ab\)

• All factors: 12, \(a\), \(b\)
• Numerical factor: 12
• Variable factors: \(a\), \(b\)
• Can also be written as: \(12 \cdot a \cdot b\) or \(2 \cdot 6 \cdot a \cdot b\) or \(3 \cdot 4 \cdot a \cdot b\)

Example 2: \(7x^2y\)

• All factors: 7, \(x\), \(x\), \(y\) (or 7, \(x^2\), \(y\))
• Written as product: \(7 \cdot x \cdot x \cdot y\)
• Numerical factor: 7
• Variable factors: \(x^2\), \(y\)

Example 3: \(-3m^2n^3\)

• Numerical factor: -3
• Variable factors: \(m^2\) (or \(m \cdot m\)), \(n^3\) (or \(n \cdot n \cdot n\))
• Expanded form: \(-3 \cdot m \cdot m \cdot n \cdot n \cdot n\)

Sorting Activity:

Given the expression: \(6x^2 + 4xy - 9y^2\)

Key Difference:

Example: In \(3x + 5\):

• Terms: \(3x\) and \(5\)
• Factors of \(3x\): 3 and \(x\)

Quick Reference: Expression Components

💡 Key Tips for Working with Expressions

• ✓ Always use parentheses when substituting values into expressions
• ✓ Follow PEMDAS/BODMAS strictly when evaluating expressions
• ✓ Pay attention to key words like "sum," "difference," "product," and "quotient"
• ✓ "Less than" and "subtracted from" reverse the order of the numbers
• ✓ Absolute value always produces a non-negative result
• ✓ Simplify inside radicals and absolute values first before applying the operation
• ✓ Check for division by zero in rational expressions
• ✓ Terms are separated by + or −; factors are multiplied together
• ✓ When no coefficient is visible, it's 1 or -1
• ✓ Use commas in word problems to determine where parentheses should go