Algebra 1 · Core Practice

Core Practice
Workbook

20 Essential Problems — 5 Core Units

Concept Review · Worked Examples · Practice Problems · Answer Key

20 Problems 5 Units Free Response
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Unit 1

Variables & Expressions

📘 Concept Review

A variable is a letter that represents an unknown value. An algebraic expression combines variables, numbers, and operations — but has no equal sign. To evaluate an expression, substitute the given value and calculate.

Variable: a symbol (usually a letter) representing an unknown quantity
Coefficient: the number multiplied by a variable (e.g., 3 in 3x)
Constant: a fixed number with no variable (e.g., −9)
Like terms: terms with the same variable and exponent — they can be combined
✏️ Worked Example

Evaluate 4x + 7 when x = 3.

4(3) + 7 = 12 + 7 = 19
📝 Practice Problems
1
Evaluate the expression 5a − 2b when a = 4 and b = 3.
Answer:
2
Write an algebraic expression for "seven less than three times a number n."
Answer:
3
Identify all the terms, coefficients, and constants in the expression: 6x + 4y − 9.
Answer:
4
Simplify by combining like terms: 3x + 5 + 7x − 2.
Answer:
Unit 2

Solving One-Step Equations

📘 Concept Review

Use inverse operations to isolate the variable. Whatever you do to one side of the equation, you must do to the other side to keep it balanced.

Inverse operations: opposite operations — addition ↔ subtraction, multiplication ↔ division
Solution: the value that makes the equation true
Isolate: get the variable alone on one side
✏️ Worked Example

Solve: x + 9 = 14.

Subtract 9 from both sides → x = 5
📝 Practice Problems
5
Solve for x: x − 13 = 7.
Answer:
6
Solve for y: 4y = −28.
Answer:
7
Solve for n: n ÷ 6 = 5.
Answer:
8
Solve for x: x + (−4) = −11.
Answer:
Unit 3

Solving Two-Step Equations

📘 Concept Review

Reverse the order of operations to solve. When solving ax + b = c, first undo the addition or subtraction (dealing with b), then undo the multiplication or division (dealing with a).

Step 1: Add or subtract to isolate the term with the variable
Step 2: Multiply or divide to solve for the variable
Check: substitute your answer back to verify both sides are equal
✏️ Worked Example

Solve: 2x + 5 = 13.

2x = 8 → x = 4
📝 Practice Problems
9
Solve for x: 3x − 7 = 14.
Answer:
10
Solve for y: −2y + 10 = 4.
Answer:
11
A number is multiplied by 5 and then 8 is added, giving 43. Write and solve the equation to find the number.
Answer:
12
Solve for x: (x + 3) ÷ 4 = 5.
Answer:
Unit 4

Inequalities

📘 Concept Review

Solve inequalities just like equations, with one critical exception: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

< less than, > greater than, ≤ less than or equal, ≥ greater than or equal
Flip rule: multiplying or dividing by a negative → reverse the sign direction
Graphing: open circle for < or >; closed circle for ≤ or ≥
✏️ Worked Example

Solve: −3x < 12.

Divide by −3 and flip: x > −4
📝 Practice Problems
13
Solve and describe the graph: x + 5 > 9. What type of circle is used and in which direction does the arrow point?
Answer:
14
Solve: −5x ≥ 20.
Answer:
15
Solve: 2x − 3 < 7.
Answer:
16
Write and solve an inequality: "a number decreased by 4 is at most 10."
Answer:
Unit 5

Linear Equations & Slope

📘 Concept Review

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Slope tells you how steep the line is and which direction it goes.

Slope (m): rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
y-intercept (b): the point where the line crosses the y-axis (x = 0)
x-intercept: the point where the line crosses the x-axis (y = 0)
Point-slope form: y − y₁ = m(x − x₁)
✏️ Worked Example

Find the slope of the line through (1, 3) and (4, 9).

m = (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2
📝 Practice Problems
17
Find the slope and y-intercept of the line y = −3x + 7.
Answer:
18
Write the equation of a line with slope 2 that passes through the point (0, −5).
Answer:
19
Find the slope of the line passing through (−2, 1) and (4, 13).
Answer:
20
A line passes through (3, 5) with slope 1/2. Write its equation in slope-intercept form.
Answer:

Answer Key & Solutions

Review every step. For any missed question, re-read the concept box and try again.

Q1 — Unit 1
Answer: 14
5(4) − 2(3) = 20 − 6 = 14
Q2 — Unit 1
Answer: 3n − 7
"Three times n" = 3n; "seven less than" = subtract 7
Q3 — Unit 1
Terms: 6x, 4y, −9 | Coeff: 6, 4 | Constant: −9
Coefficients are the numbers in front of variables; −9 has no variable
Q4 — Unit 1
Answer: 10x + 3
3x + 7x = 10x; 5 − 2 = 3
Q5 — Unit 2
x = 20
Add 13 to both sides: x = 7 + 13 = 20
Q6 — Unit 2
y = −7
Divide both sides by 4: y = −28 ÷ 4 = −7
Q7 — Unit 2
n = 30
Multiply both sides by 6: n = 5 × 6 = 30
Q8 — Unit 2
x = −7
Add 4 to both sides: x = −11 + 4 = −7
Q9 — Unit 3
x = 7
Step 1: 3x = 21 (add 7). Step 2: x = 7 (÷ 3)
Q10 — Unit 3
y = 3
Step 1: −2y = −6 (subtract 10). Step 2: y = 3 (÷ −2)
Q11 — Unit 3
n = 7 (Equation: 5n + 8 = 43)
5n = 35 → n = 7
Q12 — Unit 3
x = 17
Multiply by 4: x + 3 = 20. Subtract 3: x = 17
Q13 — Unit 4
x > 4 (open circle at 4, arrow right)
Subtract 5: x > 4. "Greater than" → open circle, arrow right
Q14 — Unit 4
x ≤ −4
Divide by −5 and FLIP the sign: x ≤ −4
Q15 — Unit 4
x < 5
Add 3: 2x < 10. Divide by 2: x < 5
Q16 — Unit 4
n ≤ 14 (Inequality: n − 4 ≤ 10)
Add 4 to both sides: n ≤ 14
Q17 — Unit 5
Slope = −3; y-intercept = 7
Read directly from y = mx + b form
Q18 — Unit 5
y = 2x − 5
(0, −5) is the y-intercept → b = −5; m = 2
Q19 — Unit 5
m = 2
(13 − 1) ÷ (4 − (−2)) = 12 ÷ 6 = 2
Q20 — Unit 5
y = (1/2)x + 7/2
5 = (1/2)(3) + b → b = 5 − 3/2 = 7/2