Algebra Quick Review Sheet
Algebra 1 • Guided Review
Quarter 1 Review: Expressions
Core Skills: Translating Expressions • Order of Operations • Combining Like Terms
Name:
Date: Period:
Skill 1: Translating Expressions
Study Guide Rule: Rate words (per, each) multiply the variable \(x\). Flat fees are one-time constants added once.
Total = (Rate \(\cdot x\)) + Flat Fee
Problem 1: Word Problem Translation Multiple Choice
An online store sells birdhouses for $34.95 each. For each shipment, there is a flat one-time shipping fee of $7.50. Which expression represents the total cost to order \(x\) birdhouses?
(A) \(x + 34.95 + 7.50\)
(B) \((34.95 + 7.50)x\)
(C) \(7.50x + 34.95\)
(D) \(34.95x + 7.50\)
Hint: Which cost repeats for every birdhouse bought?
Correct Choice:
Skill 2: Operations & Like Terms
Problem 2: Order of Ops PEMDAS
Simplify the expression completely:
\[\frac{30 \div 5 \times 2}{2 + 3^2}\]
Guide: Top: Multiply & divide left-to-right. Bottom: Evaluate exponent \(3^2\) first, then add.
Final Value:
Problem 3: Distribute & Combine Like Terms
Simplify by distributing the \(-3\) and combining like terms:
\(-3(x - 4) + x\)
Guide: Multiply \(-3 \cdot x\) and \(-3 \cdot (-4)\). Remember: a negative times a negative is positive!
Simplified Form:
Algebra 1 • Simplified Review Sheet Page 1 of 3
Part 2
Linear & Literal Equations
Q1 IA Review • Scaffolded
Skill 3: Determining Number of Solutions
1 Solution \(x = a\) Variables remain; single exact value.
0 Solutions (No Sol.) \(-10 = 24\) (False) Variables cancel out; numbers unequal.
Infinitely Many \(12 = 12\) (True) Variables cancel out; both sides identical.
Problem 4: Classifying Solutions Step-by-Step
How many solutions does the given equation have?
\(-2(5 - 9x) = 6(3x + 4)\)
Distribute both sides & compare:
Select your conclusion:
(A) Exactly one
(B) Exactly two
(C) Infinitely many
(D) Zero (No solution)
Skill 4: Solving Complex & Literal Equations
Problem 5: Fraction Equation Solve for \(x\)
Solve for \(x\). Show all algebraic steps:
\(\frac{4x + 6}{3} = 8\)
Hint: Multiply both sides by \(3\) first.
Answer:
\(x =\)
Problem 6: Literal Equation Isolate Variable
Rearrange to express \(a\) in terms of \(b\) and \(c\):
\(9a - 2b = c\)
Step 1: Add \(2b\) \(\to\) Step 2: Divide by \(9\)
Answer:
\(a =\)
Algebra 1 • Simplified Review Sheet Page 2 of 3
Part 3
Functions & Graph Analysis
Q1 IA Review • Scaffolded
Skill 5: Evaluating Functions (Equation vs. Graph)
Problem 7: Function Equation Input vs Output
The function \(f\) is defined by \(f(x) = 3x + 12\).
Part A: When is \(f(x) = 27\)?
Set output to 27 and solve for \(x\): \(3x + 12 = 27\)
\(x =\) ______
Part B: Find \(f(4)\).
Substitute \(x = 4\) into the rule:
\(f(4) =\) ______
Problem 8: Read the Graph Curve Points
Use the curve of \(y = f(x)\) below to answer:
x y -2 2 2 -2
Find \(f(-1)\):
Find \(f(3)\):
When \(f(x) = -1\), \(x =\):
Skill 6: Domain & Range from Graphs
Domain \((x)\): Horizontal span from Left \(\longleftrightarrow\) Right.
Range \((y)\): Vertical span from Bottom \(\updownarrow\) Top.
Problem 9 x y -2 2 2 -3
Part A: Domain of the function
Starts at \(x = -2\) (solid dot) and continues right \(\to\)
(A) \(x > -2\)
(B) \(x \ge -2\)
(C) \(x \le -2\)
(D) All Real Numbers
Part B: Range of the function
Starts at \(y = -3\) (lowest point) and continues upward \(\uparrow\)
(A) \(y \ge -3\)
(B) \(y > -3\)
(C) \(y \le -3\)
(D) \(y \ge 0\)
Algebra 1 • Simplified Review Sheet Page 3 of 3
Algebra Quick Review Key
Teacher Resource Complete Solutions
Quarter 1 Review Key: Expressions
Page 1 Solutions • Step-by-Step Rationale & Student Pitfalls
Answer Key
Algebra 1 Core
Skill 1: Translating Expressions — Solutions
Problem 1: Birdhouse Order Cost Answer: (D) \(34.95x + 7.50\)
Step-by-Step Breakdown:
- Rate per birdhouse = $34.95 \(\implies 34.95x\) (depends on quantity \(x\)).
- One-time shipping & handling = $7.50 (constant fee, paid once).
- Combine components: \(\text{Total} = 34.95x + 7.50\).
Common Pitfall: Students choosing (B) add \(34.95 + 7.50 = 42.45\) and multiply by \(x\), failing to recognize that shipping is charged only once per order, not per item.
Skill 2: Operations & Like Terms — Solutions
Problem 2: Order of Ops \(\frac{12}{11}\) (or \(1 \frac{1}{11}\))
Numerator (Left-to-Right): \(30 \div 5 \times 2 = 6 \times 2 = \mathbf{12}\)
Denominator (Exponents first): \(2 + 3^2 = 2 + 9 = \mathbf{11}\)
Value \(= \frac{12}{11} \approx 1.09\)
Watch out: Do not do \(5 \times 2 = 10\) first. Multiplication and division have equal priority and must be done left to right.
Problem 3: Distribute & Combine \(-2x + 12\)
Step 1: Distribute \(-3\): \(-3(x) + (-3)(-4) = \mathbf{-3x + 12}\)
Step 2: Combine like terms with \(+x\): \(-3x + x + 12 = \mathbf{-2x + 12}\)
Watch out: \(-3 \times -4 = +12\). Many students write \(-3x - 12\), losing the negative times negative rule.
Algebra 1 • Teacher Answer Key Page 1 of 3
Teacher Key • Part 2
Linear Equations & Formulas Solutions
Q1 IA Review Key
Skill 3: Number of Solutions — Solutions
Problem 4: \(-2(5 - 9x) = 6(3x + 4)\) Answer: (D) Zero (No Solution)
1. Distribute both sides:\(-10 + 18x = 18x + 24\)
2. Subtract \(18x\) from both sides:\(-10 = 24\)
\(-10 = 24\) is a FALSE statement (contradiction).
The variable drops out and the remaining numbers are unequal. Therefore, no value of \(x\) can satisfy the equation.
Key Concept: Students frequently confuse No Solution with Infinitely Many.
• False (e.g. \(-10 = 24\)) \(\to\) Zero
• True (e.g. \(24 = 24\)) \(\to\) Infinite
Skill 4: Fractional & Literal Equations — Solutions
