Expressions - Ninth Grade Math
1. Write Variable Expressions
Variable: A letter (like $x$, $y$, $z$) that represents an unknown number
Constant: A fixed number (like 5, -3, 0.5)
Coefficient: The number multiplied by a variable (in $5x$, the coefficient is 5)
Expression: A combination of variables, numbers, and operations
Constant: A fixed number (like 5, -3, 0.5)
Coefficient: The number multiplied by a variable (in $5x$, the coefficient is 5)
Expression: A combination of variables, numbers, and operations
Translating Words to Algebraic Expressions
| Operation | Key Words | Example Phrase | Expression |
|---|---|---|---|
| Addition (+) | sum, plus, more than, increased by, total, added to | "5 more than a number" | $x + 5$ |
| Subtraction (−) | difference, minus, less than, decreased by, subtracted from | "8 less than a number" | $x - 8$ |
| Multiplication (×) | product, times, multiplied by, of, twice | "3 times a number" | $3x$ or $3 \cdot x$ |
| Division (÷) | quotient, divided by, ratio, per | "a number divided by 4" | $\frac{x}{4}$ or $x \div 4$ |
Examples of Writing Variable Expressions:
1. "The sum of a number and 7" → $x + 7$
2. "6 less than twice a number" → $2x - 6$
3. "The product of 5 and a number, increased by 3" → $5x + 3$
4. "Half of a number" → $\frac{x}{2}$ or $\frac{1}{2}x$
5. "The quotient of a number and 9" → $\frac{x}{9}$
6. "3 more than the product of 4 and a number" → $4x + 3$
7. "A number decreased by 10" → $x - 10$
8. "Triple a number, minus 5" → $3x - 5$
1. "The sum of a number and 7" → $x + 7$
2. "6 less than twice a number" → $2x - 6$
3. "The product of 5 and a number, increased by 3" → $5x + 3$
4. "Half of a number" → $\frac{x}{2}$ or $\frac{1}{2}x$
5. "The quotient of a number and 9" → $\frac{x}{9}$
6. "3 more than the product of 4 and a number" → $4x + 3$
7. "A number decreased by 10" → $x - 10$
8. "Triple a number, minus 5" → $3x - 5$
Important Note on Order:
• For subtraction: "5 less than $x$" = $x - 5$ (NOT $5 - x$)
• For division: "$x$ divided by 3" = $\frac{x}{3}$ (NOT $\frac{3}{x}$)
• Order matters for subtraction and division!
• For subtraction: "5 less than $x$" = $x - 5$ (NOT $5 - x$)
• For division: "$x$ divided by 3" = $\frac{x}{3}$ (NOT $\frac{3}{x}$)
• Order matters for subtraction and division!
2. Evaluate Variable Expressions Involving Integers
To Evaluate an Expression:
1. Substitute the given value(s) for the variable(s)
2. Follow the order of operations (PEMDAS/BODMAS)
3. Simplify to get a single numerical answer
1. Substitute the given value(s) for the variable(s)
2. Follow the order of operations (PEMDAS/BODMAS)
3. Simplify to get a single numerical answer
Order of Operations (PEMDAS):
Parentheses: $( )$, $[ ]$, $\{ \}$
Exponents: Powers and roots
Multiplication and Division: Left to right
Addition and Subtraction: Left to right
Parentheses: $( )$, $[ ]$, $\{ \}$
Exponents: Powers and roots
Multiplication and Division: Left to right
Addition and Subtraction: Left to right
Example 1: Evaluate $3x + 7$ when $x = 4$
Step 1: Substitute: $3(4) + 7$
Step 2: Multiply: $12 + 7$
Step 3: Add: $19$
Answer: 19
Step 1: Substitute: $3(4) + 7$
Step 2: Multiply: $12 + 7$
Step 3: Add: $19$
Answer: 19
Example 2: Evaluate $2x - 5y$ when $x = -3$ and $y = 4$
Step 1: Substitute: $2(-3) - 5(4)$
Step 2: Multiply: $-6 - 20$
Step 3: Subtract: $-26$
Answer: -26
Step 1: Substitute: $2(-3) - 5(4)$
Step 2: Multiply: $-6 - 20$
Step 3: Subtract: $-26$
Answer: -26
Example 3: Evaluate $x^2 - 3x + 5$ when $x = -2$
Step 1: Substitute: $(-2)^2 - 3(-2) + 5$
Step 2: Exponents: $4 - 3(-2) + 5$
Step 3: Multiply: $4 + 6 + 5$
Step 4: Add: $15$
Answer: 15
Step 1: Substitute: $(-2)^2 - 3(-2) + 5$
Step 2: Exponents: $4 - 3(-2) + 5$
Step 3: Multiply: $4 + 6 + 5$
Step 4: Add: $15$
Answer: 15
Remember:
• When substituting negative numbers, use parentheses: $(-3)$
• $(-x)^2 = x^2$ (negative squared is positive)
• Always follow PEMDAS strictly
• When substituting negative numbers, use parentheses: $(-3)$
• $(-x)^2 = x^2$ (negative squared is positive)
• Always follow PEMDAS strictly
3. Evaluate Variable Expressions Involving Rational Numbers
Rational Numbers Include:
• Integers: $-3, 0, 5$
• Fractions: $\frac{1}{2}, \frac{-3}{4}, \frac{5}{8}$
• Decimals: $0.5, -2.75, 3.125$
• Mixed numbers: $2\frac{1}{3}, -1\frac{1}{2}$
• Integers: $-3, 0, 5$
• Fractions: $\frac{1}{2}, \frac{-3}{4}, \frac{5}{8}$
• Decimals: $0.5, -2.75, 3.125$
• Mixed numbers: $2\frac{1}{3}, -1\frac{1}{2}$
Example 1: Evaluate $\frac{2}{3}x + 5$ when $x = 6$
Step 1: Substitute: $\frac{2}{3}(6) + 5$
Step 2: Multiply: $\frac{12}{3} + 5 = 4 + 5$
Step 3: Add: $9$
Answer: 9
Step 1: Substitute: $\frac{2}{3}(6) + 5$
Step 2: Multiply: $\frac{12}{3} + 5 = 4 + 5$
Step 3: Add: $9$
Answer: 9
Example 2: Evaluate $4x - 2y$ when $x = 0.5$ and $y = 1.25$
Step 1: Substitute: $4(0.5) - 2(1.25)$
Step 2: Multiply: $2 - 2.5$
Step 3: Subtract: $-0.5$
Answer: -0.5 or $-\frac{1}{2}$
Step 1: Substitute: $4(0.5) - 2(1.25)$
Step 2: Multiply: $2 - 2.5$
Step 3: Subtract: $-0.5$
Answer: -0.5 or $-\frac{1}{2}$
Example 3: Evaluate $\frac{x + y}{2}$ when $x = \frac{3}{4}$ and $y = \frac{1}{2}$
Step 1: Substitute: $\frac{\frac{3}{4} + \frac{1}{2}}{2}$
Step 2: Add fractions (numerator): $\frac{\frac{3}{4} + \frac{2}{4}}{2} = \frac{\frac{5}{4}}{2}$
Step 3: Divide: $\frac{5}{4} \div 2 = \frac{5}{4} \times \frac{1}{2} = \frac{5}{8}$
Answer: $\frac{5}{8}$
Step 1: Substitute: $\frac{\frac{3}{4} + \frac{1}{2}}{2}$
Step 2: Add fractions (numerator): $\frac{\frac{3}{4} + \frac{2}{4}}{2} = \frac{\frac{5}{4}}{2}$
Step 3: Divide: $\frac{5}{4} \div 2 = \frac{5}{4} \times \frac{1}{2} = \frac{5}{8}$
Answer: $\frac{5}{8}$
4. Evaluate Rational Expressions
Rational Expression: A fraction where the numerator and/or denominator contains algebraic expressions
General Form: $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$
General Form: $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$
Steps to Evaluate Rational Expressions:
1. Substitute the given value(s) into the expression
2. Evaluate the numerator separately
3. Evaluate the denominator separately
4. Divide numerator by denominator
5. Simplify if possible
1. Substitute the given value(s) into the expression
2. Evaluate the numerator separately
3. Evaluate the denominator separately
4. Divide numerator by denominator
5. Simplify if possible
Example 1: Evaluate $\frac{2x + 3}{x - 1}$ when $x = 4$
Step 1: Substitute: $\frac{2(4) + 3}{4 - 1}$
Step 2: Evaluate numerator: $\frac{8 + 3}{4 - 1} = \frac{11}{3}$
Step 3: Evaluate denominator: $\frac{11}{3}$
Answer: $\frac{11}{3}$ or $3\frac{2}{3}$
Step 1: Substitute: $\frac{2(4) + 3}{4 - 1}$
Step 2: Evaluate numerator: $\frac{8 + 3}{4 - 1} = \frac{11}{3}$
Step 3: Evaluate denominator: $\frac{11}{3}$
Answer: $\frac{11}{3}$ or $3\frac{2}{3}$
Example 2: Evaluate $\frac{x^2 - 4}{x + 2}$ when $x = 3$
Step 1: Substitute: $\frac{3^2 - 4}{3 + 2}$
Step 2: Evaluate: $\frac{9 - 4}{5} = \frac{5}{5}$
Step 3: Simplify: $1$
Answer: 1
Step 1: Substitute: $\frac{3^2 - 4}{3 + 2}$
Step 2: Evaluate: $\frac{9 - 4}{5} = \frac{5}{5}$
Step 3: Simplify: $1$
Answer: 1
Important: A rational expression is undefined when the denominator equals zero.
Example: $\frac{x + 1}{x - 3}$ is undefined when $x = 3$
Example: $\frac{x + 1}{x - 3}$ is undefined when $x = 3$
5. Properties of Addition and Multiplication
Commutative Property
Addition: $a + b = b + a$
Multiplication: $a \cdot b = b \cdot a$
Order doesn't matter when adding or multiplying
Multiplication: $a \cdot b = b \cdot a$
Order doesn't matter when adding or multiplying
Examples:
• $3 + 5 = 5 + 3 = 8$
• $2 \times 7 = 7 \times 2 = 14$
• $x + 4 = 4 + x$
• $5y = y \cdot 5$
• $3 + 5 = 5 + 3 = 8$
• $2 \times 7 = 7 \times 2 = 14$
• $x + 4 = 4 + x$
• $5y = y \cdot 5$
Associative Property
Addition: $(a + b) + c = a + (b + c)$
Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Grouping doesn't matter when adding or multiplying
Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Grouping doesn't matter when adding or multiplying
Examples:
• $(2 + 3) + 4 = 2 + (3 + 4) = 9$
• $(2 \times 3) \times 5 = 2 \times (3 \times 5) = 30$
• $(x + 2) + 5 = x + (2 + 5) = x + 7$
• $(3x) \cdot 4 = 3 \cdot (x \cdot 4) = 12x$
• $(2 + 3) + 4 = 2 + (3 + 4) = 9$
• $(2 \times 3) \times 5 = 2 \times (3 \times 5) = 30$
• $(x + 2) + 5 = x + (2 + 5) = x + 7$
• $(3x) \cdot 4 = 3 \cdot (x \cdot 4) = 12x$
Identity Property
Additive Identity: $a + 0 = a$
Multiplicative Identity: $a \cdot 1 = a$
Adding 0 or multiplying by 1 doesn't change the value
Multiplicative Identity: $a \cdot 1 = a$
Adding 0 or multiplying by 1 doesn't change the value
Examples:
• $7 + 0 = 7$
• $x + 0 = x$
• $9 \times 1 = 9$
• $1 \cdot y = y$
• $7 + 0 = 7$
• $x + 0 = x$
• $9 \times 1 = 9$
• $1 \cdot y = y$
Inverse Property
Additive Inverse: $a + (-a) = 0$
Multiplicative Inverse: $a \cdot \frac{1}{a} = 1$ (where $a \neq 0$)
Every number has an opposite and a reciprocal
Multiplicative Inverse: $a \cdot \frac{1}{a} = 1$ (where $a \neq 0$)
Every number has an opposite and a reciprocal
Examples:
• $5 + (-5) = 0$
• $x + (-x) = 0$
• $3 \times \frac{1}{3} = 1$
• $x \cdot \frac{1}{x} = 1$ (when $x \neq 0$)
• $5 + (-5) = 0$
• $x + (-x) = 0$
• $3 \times \frac{1}{3} = 1$
• $x \cdot \frac{1}{x} = 1$ (when $x \neq 0$)
Property Summary Table
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | $a + b = b + a$ | $a \cdot b = b \cdot a$ |
| Associative | $(a + b) + c = a + (b + c)$ | $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ |
| Identity | $a + 0 = a$ | $a \cdot 1 = a$ |
| Inverse | $a + (-a) = 0$ | $a \cdot \frac{1}{a} = 1$ ($a \neq 0$) |
Important: Subtraction and division do NOT have commutative or associative properties!
• $5 - 3 \neq 3 - 5$
• $10 \div 2 \neq 2 \div 10$
• $5 - 3 \neq 3 - 5$
• $10 \div 2 \neq 2 \div 10$
6. Distributive Property
Distributive Property Formula:
$a(b + c) = ab + ac$
$a(b - c) = ab - ac$
Multiply the outside term by each term inside the parentheses
$a(b + c) = ab + ac$
$a(b - c) = ab - ac$
Multiply the outside term by each term inside the parentheses
Expanding with Distributive Property
Example 1: $3(x + 5)$
$= 3 \cdot x + 3 \cdot 5$
$= 3x + 15$
$= 3 \cdot x + 3 \cdot 5$
$= 3x + 15$
Example 2: $-2(4x - 7)$
$= -2 \cdot 4x + (-2) \cdot (-7)$
$= -8x + 14$
$= -2 \cdot 4x + (-2) \cdot (-7)$
$= -8x + 14$
Example 3: $5(2a + 3b - 4)$
$= 5 \cdot 2a + 5 \cdot 3b + 5 \cdot (-4)$
$= 10a + 15b - 20$
$= 5 \cdot 2a + 5 \cdot 3b + 5 \cdot (-4)$
$= 10a + 15b - 20$
Example 4: $\frac{1}{2}(6x - 8)$
$= \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 8$
$= 3x - 4$
$= \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 8$
$= 3x - 4$
Factoring Using Distributive Property (Reverse)
Factoring: Find the Greatest Common Factor (GCF) and factor it out
$ab + ac = a(b + c)$
$ab + ac = a(b + c)$
Example 1: Factor $6x + 9$
GCF = 3
$= 3(2x + 3)$
GCF = 3
$= 3(2x + 3)$
Example 2: Factor $4x^2 - 8x$
GCF = $4x$
$= 4x(x - 2)$
GCF = $4x$
$= 4x(x - 2)$
Key Points:
• Distribute to ALL terms inside parentheses
• Be careful with negative signs: $-(x + 3) = -x - 3$
• When factoring, the GCF goes outside the parentheses
• Distribute to ALL terms inside parentheses
• Be careful with negative signs: $-(x + 3) = -x - 3$
• When factoring, the GCF goes outside the parentheses
7. Simplify Linear Expressions Using Properties
Linear Expression: An algebraic expression where the highest power of the variable is 1
Examples: $3x + 5$, $2x - 7$, $4x + 3y - 8$
Examples: $3x + 5$, $2x - 7$, $4x + 3y - 8$
Steps to Simplify:
1. Use distributive property to remove parentheses
2. Combine like terms
3. Write in standard form (usually with terms in descending order)
1. Use distributive property to remove parentheses
2. Combine like terms
3. Write in standard form (usually with terms in descending order)
Example 1: Simplify $2(x + 3) + 4x$
Step 1: Distribute: $2x + 6 + 4x$
Step 2: Combine like terms: $6x + 6$
Answer: $6x + 6$
Step 1: Distribute: $2x + 6 + 4x$
Step 2: Combine like terms: $6x + 6$
Answer: $6x + 6$
Example 2: Simplify $5(2x - 1) - 3(x + 4)$
Step 1: Distribute: $10x - 5 - 3x - 12$
Step 2: Combine like terms: $7x - 17$
Answer: $7x - 17$
Step 1: Distribute: $10x - 5 - 3x - 12$
Step 2: Combine like terms: $7x - 17$
Answer: $7x - 17$
Example 3: Simplify $\frac{1}{2}(6x + 4) + \frac{1}{3}(9x - 6)$
Step 1: Distribute: $3x + 2 + 3x - 2$
Step 2: Combine like terms: $6x$
Answer: $6x$
Step 1: Distribute: $3x + 2 + 3x - 2$
Step 2: Combine like terms: $6x$
Answer: $6x$
8. Sort Factors of Variable Expressions
Factor: A number or variable that is multiplied to form a product
Term: A single number, variable, or product of numbers and variables
Coefficient: The numerical factor in a term
Term: A single number, variable, or product of numbers and variables
Coefficient: The numerical factor in a term
Understanding Factors vs Terms
Factors are multiplied: In $5xy$, the factors are $5$, $x$, and $y$
Terms are added/subtracted: In $5xy + 3x - 7$, the terms are $5xy$, $3x$, and $-7$
Terms are added/subtracted: In $5xy + 3x - 7$, the terms are $5xy$, $3x$, and $-7$
Example 1: Identify factors of $12x^2y$
Factors: $1, 2, 3, 4, 6, 12, x, x^2, y, xy, x^2y, 2x, 3x, 4x, 6x, 12x$, etc.
Prime factorization: $2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$
Factors: $1, 2, 3, 4, 6, 12, x, x^2, y, xy, x^2y, 2x, 3x, 4x, 6x, 12x$, etc.
Prime factorization: $2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$
Example 2: List factors of the expression $3x^2 + 6x$
Terms: $3x^2$ and $6x$
Factors of $3x^2$: $3, x, x, x^2$ or $3 \cdot x \cdot x$
Factors of $6x$: $6, x$ or $2 \cdot 3 \cdot x$
Common factors: $3, x, 3x$ → GCF is $3x$
Terms: $3x^2$ and $6x$
Factors of $3x^2$: $3, x, x, x^2$ or $3 \cdot x \cdot x$
Factors of $6x$: $6, x$ or $2 \cdot 3 \cdot x$
Common factors: $3, x, 3x$ → GCF is $3x$
| Expression | Terms (separated by + or −) | Factors (of each term) |
|---|---|---|
| $7x$ | $7x$ | $7, x$ |
| $5xy$ | $5xy$ | $5, x, y$ |
| $3x + 4$ | $3x$ and $4$ | $(3, x)$ and $(4)$ |
| $2x^2 - 5x + 3$ | $2x^2$, $-5x$, and $3$ | $(2, x, x)$, $(5, x)$, and $(3)$ |
9. Simplify Variable Expressions Involving Like Terms and Distributive Property
Like Terms: Terms that have the same variable(s) raised to the same power(s)
Examples: $3x$ and $7x$ are like terms
$2x^2$ and $5x^2$ are like terms
BUT $3x$ and $3x^2$ are NOT like terms
Examples: $3x$ and $7x$ are like terms
$2x^2$ and $5x^2$ are like terms
BUT $3x$ and $3x^2$ are NOT like terms
Identifying Like Terms
| Like Terms | NOT Like Terms |
|---|---|
| $5x, 3x, -2x, x$ | $5x, 5x^2$ (different powers) |
| $7y^2, -3y^2, y^2$ | $7y^2, 7y$ (different powers) |
| $2xy, -5xy, xy$ | $2xy, 2x$ (different variables) |
| $8, -3, 15$ (constants) | $8, 8x$ (one has variable) |
Combining Like Terms
Rule: Add or subtract the coefficients, keep the variable part the same
$ax + bx = (a + b)x$
$ax + bx = (a + b)x$
Simple Examples:
• $5x + 3x = 8x$
• $7y - 2y = 5y$
• $4a + 3b + 2a = 6a + 3b$
• $3x^2 + 5x + 2x^2 - 3x = 5x^2 + 2x$
• $5x + 3x = 8x$
• $7y - 2y = 5y$
• $4a + 3b + 2a = 6a + 3b$
• $3x^2 + 5x + 2x^2 - 3x = 5x^2 + 2x$
Using Distributive Property and Combining Like Terms
Example 1: Simplify $3(x + 4) + 2(x - 1)$
Step 1: Distribute: $3x + 12 + 2x - 2$
Step 2: Combine like terms: $5x + 10$
Answer: $5x + 10$
Step 1: Distribute: $3x + 12 + 2x - 2$
Step 2: Combine like terms: $5x + 10$
Answer: $5x + 10$
Example 2: Simplify $5(2x + 3) - 4(x - 2) + 7$
Step 1: Distribute: $10x + 15 - 4x + 8 + 7$
Step 2: Combine like terms ($x$ terms): $10x - 4x = 6x$
Step 3: Combine constants: $15 + 8 + 7 = 30$
Answer: $6x + 30$
Step 1: Distribute: $10x + 15 - 4x + 8 + 7$
Step 2: Combine like terms ($x$ terms): $10x - 4x = 6x$
Step 3: Combine constants: $15 + 8 + 7 = 30$
Answer: $6x + 30$
Example 3: Simplify $-2(3x - 5) + 4x - (2x + 3)$
Step 1: Distribute: $-6x + 10 + 4x - 2x - 3$
Step 2: Combine like terms:
$x$ terms: $-6x + 4x - 2x = -4x$
Constants: $10 - 3 = 7$
Answer: $-4x + 7$
Step 1: Distribute: $-6x + 10 + 4x - 2x - 3$
Step 2: Combine like terms:
$x$ terms: $-6x + 4x - 2x = -4x$
Constants: $10 - 3 = 7$
Answer: $-4x + 7$
Key Steps:
1. Remove parentheses using distributive property
2. Identify like terms (same variable and exponent)
3. Add or subtract coefficients of like terms
4. Write final answer in simplified form
1. Remove parentheses using distributive property
2. Identify like terms (same variable and exponent)
3. Add or subtract coefficients of like terms
4. Write final answer in simplified form
10. Identify Equivalent Linear Expressions
Equivalent Expressions: Different-looking expressions that have the same value for all values of the variable(s)
Example: $2(x + 3)$ and $2x + 6$ are equivalent
Example: $2(x + 3)$ and $2x + 6$ are equivalent
Methods to Determine if Expressions are Equivalent
Method 1: Simplify Both Expressions
If both expressions simplify to the same form, they are equivalent
If both expressions simplify to the same form, they are equivalent
Example: Are $3(x + 2) + 4$ and $3x + 10$ equivalent?
Simplify first expression: $3x + 6 + 4 = 3x + 10$ ✓
Second expression: $3x + 10$ ✓
Yes, they are equivalent!
Simplify first expression: $3x + 6 + 4 = 3x + 10$ ✓
Second expression: $3x + 10$ ✓
Yes, they are equivalent!
Method 2: Substitute Values
Test with specific values. If they give the same result for all tested values, they're likely equivalent
(But you must simplify to be certain!)
Test with specific values. If they give the same result for all tested values, they're likely equivalent
(But you must simplify to be certain!)
Example: Are $2(x + 1) + 3x$ and $5x + 2$ equivalent?
Test with $x = 0$:
First: $2(0 + 1) + 3(0) = 2$
Second: $5(0) + 2 = 2$ ✓
Test with $x = 1$:
First: $2(1 + 1) + 3(1) = 4 + 3 = 7$
Second: $5(1) + 2 = 7$ ✓
Verify by simplifying: $2x + 2 + 3x = 5x + 2$ ✓
Test with $x = 0$:
First: $2(0 + 1) + 3(0) = 2$
Second: $5(0) + 2 = 2$ ✓
Test with $x = 1$:
First: $2(1 + 1) + 3(1) = 4 + 3 = 7$
Second: $5(1) + 2 = 7$ ✓
Verify by simplifying: $2x + 2 + 3x = 5x + 2$ ✓
Practice Examples
Which expressions are equivalent to $4x + 12$?
a) $4(x + 3)$ → Simplify: $4x + 12$ ✓ Equivalent
b) $2(2x + 6)$ → Simplify: $4x + 12$ ✓ Equivalent
c) $4x + 3 \cdot 4$ → Simplify: $4x + 12$ ✓ Equivalent
d) $2(2x + 5) + 2$ → Simplify: $4x + 10 + 2 = 4x + 12$ ✓ Equivalent
e) $4(x + 4)$ → Simplify: $4x + 16$ ✗ NOT Equivalent
a) $4(x + 3)$ → Simplify: $4x + 12$ ✓ Equivalent
b) $2(2x + 6)$ → Simplify: $4x + 12$ ✓ Equivalent
c) $4x + 3 \cdot 4$ → Simplify: $4x + 12$ ✓ Equivalent
d) $2(2x + 5) + 2$ → Simplify: $4x + 10 + 2 = 4x + 12$ ✓ Equivalent
e) $4(x + 4)$ → Simplify: $4x + 16$ ✗ NOT Equivalent
Match Equivalent Expressions:
| Expression A | Expression B | Equivalent? |
|---|---|---|
| $5(x + 2)$ | $5x + 10$ | ✓ Yes |
| $3x + 2x - 5$ | $5x - 5$ | ✓ Yes |
| $2(x - 3) + 4$ | $2x - 2$ | ✓ Yes ($2x - 6 + 4$) |
| $7x - 3x + 8$ | $4x + 8$ | ✓ Yes |
| $-(x + 5)$ | $-x - 5$ | ✓ Yes |
To Check if Expressions are Equivalent:
1. Simplify both expressions completely
2. If simplified forms match exactly → They are equivalent
3. If simplified forms differ → They are NOT equivalent
4. You can also substitute values, but simplification is the most reliable method
1. Simplify both expressions completely
2. If simplified forms match exactly → They are equivalent
3. If simplified forms differ → They are NOT equivalent
4. You can also substitute values, but simplification is the most reliable method
Quick Reference Guide
Key Vocabulary:
• Variable: A letter representing an unknown ($x, y, z$)
• Constant: A fixed number (5, -3, 0.5)
• Coefficient: Number in front of variable ($5$ in $5x$)
• Term: Single part separated by + or − ($3x$, $-7$, $4y^2$)
• Like Terms: Same variable(s) and power(s) ($3x$ and $7x$)
• Expression: Combination of terms ($3x + 5$)
• Factor: Numbers/variables multiplied together
• Variable: A letter representing an unknown ($x, y, z$)
• Constant: A fixed number (5, -3, 0.5)
• Coefficient: Number in front of variable ($5$ in $5x$)
• Term: Single part separated by + or − ($3x$, $-7$, $4y^2$)
• Like Terms: Same variable(s) and power(s) ($3x$ and $7x$)
• Expression: Combination of terms ($3x + 5$)
• Factor: Numbers/variables multiplied together
Essential Formulas:
• Distributive Property: $a(b + c) = ab + ac$
• Combining Like Terms: $ax + bx = (a+b)x$
• Commutative (Addition): $a + b = b + a$
• Commutative (Multiplication): $ab = ba$
• Associative (Addition): $(a+b)+c = a+(b+c)$
• Associative (Multiplication): $(ab)c = a(bc)$
• Identity (Addition): $a + 0 = a$
• Identity (Multiplication): $a \cdot 1 = a$
• Distributive Property: $a(b + c) = ab + ac$
• Combining Like Terms: $ax + bx = (a+b)x$
• Commutative (Addition): $a + b = b + a$
• Commutative (Multiplication): $ab = ba$
• Associative (Addition): $(a+b)+c = a+(b+c)$
• Associative (Multiplication): $(ab)c = a(bc)$
• Identity (Addition): $a + 0 = a$
• Identity (Multiplication): $a \cdot 1 = a$
Simplification Process:
Step 1: Remove parentheses (use distributive property)
Step 2: Identify like terms
Step 3: Combine like terms (add/subtract coefficients)
Step 4: Write in standard form
Step 1: Remove parentheses (use distributive property)
Step 2: Identify like terms
Step 3: Combine like terms (add/subtract coefficients)
Step 4: Write in standard form
Evaluation Process:
Step 1: Substitute given values for variables
Step 2: Follow PEMDAS order of operations
Step 3: Simplify to get numerical answer
Step 1: Substitute given values for variables
Step 2: Follow PEMDAS order of operations
Step 3: Simplify to get numerical answer
Study Tips:
✓ Always check your work by substituting values
✓ Be careful with negative signs when distributing
✓ Combine only like terms (same variable and power)
✓ Remember: order matters for subtraction and division!
✓ Parentheses are crucial when substituting negative values
✓ Always check your work by substituting values
✓ Be careful with negative signs when distributing
✓ Combine only like terms (same variable and power)
✓ Remember: order matters for subtraction and division!
✓ Parentheses are crucial when substituting negative values