Compound Inequalities Bonus Worksheet ALGEBRA 1 • UNIT 1 EXTENSION QUIZ BONUS CHALLENGE PAGE 1 OF 1
Compound Inequalities
Advanced Modeling • Bonus Problems
Name:
Date:
Period:
Bonus Instructions: Define variables, set up algebraic compound inequalities, and show every inverse operation.
Earn up to +10 Points
1
Find three consecutive odd integers such that five less than twice their sum is strictly between 61 and 85.
+4 PTS
1. Define Variables
Let 1st odd int =
Let 2nd odd int =
Let 3rd odd int =
Sum in terms of n: _______
2. Write & Solve Inequality
Range for n: ____________
3. The Three Integers
{ , , }
Check: 2(Sum) − 5 =
2
Five more than twice a number is either at most −3 or at least 11.
Write a compound inequality, solve for all possible values of the number, and graph the solution set.
+3 PTS
Variable:
Compound Inequality:
Solve "At Most −3" ≤
Branch 1: _________________________
Solve "At Least 11" ≥
Branch 2: _________________________
Final Solution: Solution:
-8 -6 -4 -2 0 2 4 6 8
3
Fourteen less than a number is between −6 and 11.
Find the range of numbers that make this statement true.
+3 PTS
Variable:
Inequality:
Answer:
Algebra 1 • Lesson 1-6 Extension Show all inverse operations clearly for full credit Page 1 of 1
Compound Inequalities Answer Key ALGEBRA 1 • UNIT 1 EXTENSION TEACHER ANSWER KEY PAGE 1 OF 1
Compound Inequalities
Full Worked Solutions • Scoring Rubric
Total Points 10 Pts Bonus
+10 Max
Grading Note: Award partial credit for correct algebraic setups even if minor arithmetic errors occur.
P1: 4 pts • P2: 3 pts • P3: 3 pts
1
Find three consecutive odd integers such that five less than twice their sum is strictly between 61 and 85.
+4 PTS
1. Variable Setup (1 pt)
1st int: n
2nd int: n + 2
3rd int: n + 4
Sum = 3n + 6
2. Algebraic Steps (2 pts)
\(61 < 2(3n + 6) - 5 < 85\) (1 pt)
\(61 < 6n + 7 < 85\)
\(54 < 6n < 78\)
\(9 < n < 13\) (1 pt)
n must be an odd integer
3. Final Solution (1 pt)
Since \(n \in (9, 13)\) is odd:
{ 11, 13, 15 }
✓ Check: 2(39) − 5 = 73 ∈ (61, 85)
2
Five more than twice a number is either at most −3 or at least 11.
Write a compound inequality, solve for all possible values of the number, and graph the solution set.
+3 PTS
Variable (0.5 pt): Let x = the number
Inequality (0.5 pt): 2x + 5 ≤ −3 or 2x + 5 ≥ 11
Left Branch (≤ −3) • 0.5 pt ≤
\(2x + 5 \le -3\)
\(x \le -4\)
Right Branch (≥ 11) • 0.5 pt ≥
\(2x + 5 \ge 11\)
\(x \ge 3\)
Solution (0.5 pt) & Graph (0.5 pt): x ≤ −4 or x ≥ 3 (−∞, −4] ∪ [3, ∞)
-8 -6 -4 -2 0 2 3 4 6 8
3
Fourteen less than a number is between −6 and 11.
Find the range of numbers that make this statement true.
+3 PTS
Variable (0.5 pt): Let n = the number
Inequality (1 pt): −6 < n − 14 < 11
Solve (0.5 pt):−6 < n − 14 < 11
Add 14:−6 + 14 < n − 14 + 14 < 11 + 14
Result:8 < n < 25
Key Misconception:
Watch for students writing \(14 - n\) instead of \(n - 14\) for "fourteen less than a number".
Answer (1 pt): 8 < n < 25 or (8, 25)
Algebra 1 • Lesson 1-6 Extension Answer Key • For Teacher Facilitation and Grading Page 1 of 1
Absolute Value Practice Worksheet ENVISION ALGEBRA 1 • LESSON 1-7 ADDITIONAL PRACTICE PAGE 1 OF 2
Absolute Value Equations
1-7 Practice with Guided Workspace
Name:
Date:
Period:
Part 1: Solve each equation. Graph and check your solutions on the number lines provided.
Problems 1–6
1 |b| = 2/3
Graph • Check
1. Solve (Two Cases):
b = ________ or b = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -2 -1 0 1 2
2 10 = |y|
Graph • Check
1. Solve (Two Cases):
y = ________ or y = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -15 -10 -5 0 5 10 15
3 |n| + 2 = 5
Isolate • Graph
1. Isolate |n| & Solve:
n = ________ or n = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -5 -3 0 3 5
4 4 = |s| − 3
Isolate • Graph
1. Isolate |s| & Solve:
s = ________ or s = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -10 -7 0 7 10
5 |x| − 5 = −1
Isolate • Graph
1. Isolate |x| & Solve:
x = ________ or x = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -6 -4 0 4 6
6 7|d| = 49
Divide • Graph
1. Divide by 7 & Solve:
d = ________ or d = ________
2. Check Solutions:
✓ Both check
3. Graph on Number Line: -10 -7 0 7 10
enVision™ Algebra 1 • Teaching Resources Remember: An absolute value cannot equal a negative number • Always check both cases Page 1 of 2
ENVISION ALGEBRA 1 • LESSON 1-7 EQUATION PRACTICE • SPECIAL CASES PAGE 2 OF 2
Solving Absolute Value Equations
Isolate, Split Cases, or Identify No Solution
Name:
Part 2: Solve each equation. If there is no solution, write no solution .
Problems 7–15
7 |r − 9| = −3
Watch Sign
Solution:
8 |c + 3| = 15
Solution:
9 1 = |g + 3|
Solution:
Absolute Value Practice Answer Key ENVISION ALGEBRA 1 • LESSON 1-7 TEACHER ANSWER KEY PAGE 1 OF 2
Absolute Value Equations
Full Worked Solutions • Problems 1–6
Key Type Complete Key
✓ Graph & Check
Part 1 Key: Students must isolate the absolute value, state both cases, check solutions, and plot discrete points.
Problems 1–6
1 |b| = 2/3
b = ±2/3
1. Two Cases:
Case 1: b = 2/3
Case 2: b = −2/3
Both are valid solutions
2. Check:
|2/3| = 2/3 ✓
|-2/3| = 2/3 ✓
✓ Verified
3. Plotted Points: −2/3 and +2/3 -2 -1 0 1 2
2 10 = |y|
y = ±10
1. Two Cases:
|y| = 10
Case 1: y = 10
Case 2: y = −10
Solution: y = −10, 10
2. Check:
|10| = 10 ✓
|-10| = 10 ✓
✓ Verified
3. Plotted Points: −10 and +10 -15 -10 -5 0 5 10 15
3 |n| + 2 = 5
n = ±3
1. Isolate & Solve:
|n| = 5 − 2 → |n| = 3
Case 1: n = 3
Case 2: n = −3
Solution: n = −3, 3
2. Check:
|3| + 2 = 5 ✓
|-3| + 2 = 5 ✓
✓ Verified
3. Plotted Points: −3 and +3 -5 -3 0 3 5
4 4 = |s| − 3
s = ±7
1. Isolate & Solve:
|s| = 4 + 3 → |s| = 7
Case 1: s = 7
Case 2: s = −7
Solution: s = −7, 7
2. Check:
|7| - 3 = 4 ✓
|-7| - 3 = 4 ✓
✓ Verified
3. Plotted Points: −7 and +7 -10 -7 0 7 10
5 |x| − 5 = −1
x = ±4
1. Isolate & Solve:
|x| = −1 + 5 → |x| = 4
Case 1: x = 4
Case 2: x = −4
Solution: x = −4, 4
2. Check:
|4| - 5 = -1 ✓
|-4| - 5 = -1 ✓
✓ Verified
3. Plotted Points: −4 and +4 -6 -4 0 4 6
Absolute Value AND Inequalities Worksheet ALGEBRA 2 • LESSON 1-7 PRACTICE WORKSHEET PAGE 1 OF 2
Absolute Value AND Inequalities
Solve Each Inequality and Graph Its Solution
Name:
Date:
Per:
Core Rule: |u| < c (c > 0) → −c < u < c (AND / bounded interval) • If |u| < (negative), solution is No Solution . Problems 1–8
1 |6n| ≤ 18
AND
Solution:
-6 -4 -2 0 2 4 6
2 |p + 4| ≤ 8
AND
Solution:
-12 -8 -4 -2 0 2 6
3 |m − 2| < 8
AND
Solution:
-8 -4 0 4 8 12
4 |5x| ≤ 10
AND
Solution:
-7 -5 -2 0 2 5
5 |x − 2| − 5 < −2
Isolate First
Solution:
-4 -2 0 2 5 8
6 |x − 4| − 3 < 5
Isolate First
Solution:
-4 0 4 8 12 14
7 1 + |1 + b| < 4
Isolate First
Solution:
-7 -4 0 2 5
8 |v + 5| − 6 < −5
Isolate First
Solution:
-10 -8 -6 -4 -2 0
Algebra 2 • Lesson 1-7 Absolute Value Inequalities Practice Turn page over for problems 9–16 Page 1 of 2
ALGEBRA 2 • LESSON 1-7 PRACTICE WORKSHEET PAGE 2 OF 2
Absolute Value AND Inequalities (Continued)
Multi-Step Operations • Special Solution Sets
Name:
Watch for negative coefficients. Check for cases where no values satisfy the inequality. Problems 9–16
9 |10p − 4| < 34
AND
Solution:
-7 -5 -3 0 3 5
10 |6 + 9x| ≤ 24
AND
Solution:
-4 -2 0 2 5 8
11 9|m − 8| − 10 < 26
2-Step Isolate
Solution:
1 4 8 12 15
12 9|x + 8| + 10 < 55
2-Step Isolate
Solution:
-16 -13 -8 -3 -2
13 9|r − 2| − 10 < −73
Watch Signs
Solution:
-6 -3 0 3 6 8
14 7|n/3| − 9 < 12
AND
Solution:
-10 -6 0 6 10
15 7 + |6v + 7| ≤ 60
AND
Solution:
-12 -8 0 4 8
16 4|6 − 2a| + 8 ≤ 24
Absolute Value AND Inequalities Key ALGEBRA 2 • LESSON 1-7 TEACHER ANSWER KEY PAGE 1 OF 2
Absolute Value AND Inequalities
Worked Solutions & Plotted Graphs • Problems 1–8
Focus: AND Inequalities (Between)
Red shaded segments represent the solution sets. Solid dots = ≤/≥; open circles = </>. Problems 1–8
1 |6n| ≤ 18
−3 ≤ n ≤ 3
−18 ≤ 6n ≤ 18
−18/6 ≤ n ≤ 18/6 → −3 ≤ n ≤ 3
Interval: [−3, 3]
-6 0 6
2 |p + 4| ≤ 8
−12 ≤ p ≤ 4
−8 ≤ p + 4 ≤ 8
−8 − 4 ≤ p ≤ 8 − 4 → −12 ≤ p ≤ 4
Interval: [−12, 4]
-12 -8 -4 0 4 6
3 |m − 2| < 8
−6 < m < 10
−8 < m − 2 < 8
−8 + 2 < m < 8 + 2 → −6 < m < 10
Interval: (−6, 10)
-8 -6 0 10 12
4 |5x| ≤ 10
−2 ≤ x ≤ 2
−10 ≤ 5x ≤ 10
−10/5 ≤ x ≤ 10/5 → −2 ≤ x ≤ 2
Interval: [−2, 2]
-7 -2 0 2 5
5 |x − 2| − 5 < −2
−1 < x < 5
Add 5: |x − 2| < 3
−3 < x − 2 < 3 → −1 < x < 5
Interval: (−1, 5)
-4 -1 0 5 8
6 |x − 4| − 3 < 5
−4 < x < 12
Add 3: |x − 4| < 8
−8 < x − 4 < 8 → −4 < x < 12
Interval: (−4, 12)
-4 0 12 14
7 1 + |1 + b| < 4
−4 < b < 2
Subtract 1: |1 + b| < 3
−3 < 1 + b < 3 → −4 < b < 2
Interval: (−4, 2)
-7 -4 0 2 5
8 |v + 5| − 6 < −5
−6 < v < −4
Add 6: |v + 5| < 1
−1 < v + 5 < 1 → −6 < v < −4
Interval: (−6, −4)
-10 -6 -4 0
Algebra 2 • Lesson 1-7 Absolute Value Inequalities Key Teacher Resource • Keep Confidential Page 1 of 2
ALGEBRA 2 • LESSON 1-7 TEACHER ANSWER KEY PAGE 2 OF 2
Absolute Value AND Inequalities (Continued)
Worked Solutions & Plotted Graphs • Problems 9–16
Focus: Multi-Step Operations • Special Cases
Problem 13 has No Solution (|r − 2| < −7). Problem 16 reverses inequality signs when dividing by −2. Problems 9–16
9 |10p − 4| < 34
−3 < p < 3.8
−34 < 10p − 4 < 34
Absolute Value OR Inequalities Worksheet ALGEBRA 2 • LESSON 1-7 PRACTICE WORKSHEET PAGE 1 OF 2
Absolute Value Inequalities
OR Inequalities & Special Solution Sets
Name:
Date:
Per:
Core Rule: |u| > c → u > c or u < −c (opposite rays) • If |u| ≥ (negative), solution is All Real Numbers . Problems 1–6
1 |x| + 5 ≥ 11
OR
Isolate |x| & write two OR cases:
Solution:
-8 -6 0 6 10
2 |m| − 2 > 0
OR
Isolate |m| & write two OR cases:
Solution:
-7 -2 0 2 7
3 |r| − 3 > 2
OR
Isolate |r| & write two OR cases:
Solution:
-8 -5 0 5 10
4 |n| + 2 ≥ 5
OR
Isolate |n| & write two OR cases:
Solution:
-6 -3 0 3 8
5 |−8a − 3| > 11
Flip on ÷ (−)
Watch out: Divide by negative reverses inequality symbol!
Solution:
-5 0 1 7
6 |1 − 4k| ≥ −11
Special Case
Recall: Is absolute value always ≥ negative numbers?
Solution:
-5 -3 0 3 7
Algebra 2 • Absolute Value Inequalities Practice Turn page over for multi-step problems 7–10 Page 1 of 2
ALGEBRA 2 • LESSON 1-7 PRACTICE WORKSHEET PAGE 2 OF 2
Multi-Step Absolute Value Inequalities
Isolate, Formulate OR Cases & Graph
Name:
Strategy: 1. Isolate the absolute value expression completely • 2. Set up both OR cases • 3. Solve & graph. Problems 7–10
7 2|10b + 7| − 1 > 73
OR
Step 1: Isolate |10b + 7|:
Case 1 (> +c):
Case 2 (< −c):
Solution:
-7 -4 0 3 6
8 9|3n − 2| + 6 > 51
OR
Step 1: Isolate |3n − 2|:
Case 1 (> +c):
Case 2 (< −c):
Solution:
-6 -1 0 2.3 6
9 3 + 4|3x + 7| ≥ −89
Special Case
Step 1: Isolate |3x + 7|:
Step 2: Analysis & Sign Check:
What does it mean when |expression| ≥ (negative number)?
Solution:
-6 -3 0 3 6
10 9|1 + 8n| − 3 ≥ 78
OR
Step 1: Isolate |1 + 8n|:
Absolute Value OR Inequalities Key ALGEBRA 2 • LESSON 1-7 TEACHER ANSWER KEY PAGE 1 OF 2
Absolute Value Inequalities
OR Inequalities & Special Solution Sets • Key (1–6)
Focus: OR Inequalities (Opposite Rays)
Red arrows indicate opposite rays. Problem 6 is an All Real Numbers special case. Problems 1–6
1 |x| + 5 ≥ 11
x ≤ −6 or x ≥ 6
Subtract 5: |x| ≥ 6
Two OR Cases: x ≥ 6 or x ≤ −6
Interval: (−∞, −6] ∪ [6, ∞)
Solution: x ≤ −6 or x ≥ 6
-8 -6 0 6 10
2 |m| − 2 > 0
m < −2 or m > 2
Add 2: |m| > 2
Two OR Cases: m > 2 or m < −2
Interval: (−∞, −2) ∪ (2, ∞)
Solution: m < −2 or m > 2
-7 -2 0 2 7
3 |r| − 3 > 2
r < −5 or r > 5
Add 3: |r| > 5
Two OR Cases: r > 5 or r < −5
Interval: (−∞, −5) ∪ (5, ∞)
Solution: r < −5 or r > 5
-8 -5 0 5 10
4 |n| + 2 ≥ 5
n ≤ −3 or n ≥ 3
Subtract 2: |n| ≥ 3
Two OR Cases: n ≥ 3 or n ≤ −3
Interval: (−∞, −3] ∪ [3, ∞)
Solution: n ≤ −3 or n ≥ 3
-6 -3 0 3 8
5 |−8a − 3| > 11
a < −1.75 or a > 1
Case 1: −8a − 3 > 11 → −8a > 14 → a < −1.75
Case 2: −8a − 3 < −11 → −8a < −8 → a > 1
Note: Dividing by −8 flips both inequalities!
Solution: a < −1.75 or a > 1
-5 -1.75 0 1 7
6 |1 − 4k| ≥ −11
All Real Numbers
Analysis: Absolute value is always ≥ 0.
Since 0 ≥ −11 is always true, |1 − 4k| ≥ −11 holds for ALL real k.
Solution: (−∞, ∞) or ℝ
Solution: All Real Numbers (ℝ)
-5 0 7
Algebra 2 • Absolute Value Inequalities Key Teacher Resource • Keep Confidential Page 1 of 2
ALGEBRA 2 • LESSON 1-7 TEACHER ANSWER KEY PAGE 2 OF 2
Multi-Step Absolute Value Inequalities
Worked Solutions & Plotted Graphs • Problems 7–10
Focus: Multi-Step OR Cases & All-Real
Problem 9 simplifies to |3x + 7| ≥ −23, yielding All Real Numbers. Problems 7–10
7 2|10b + 7| − 1 > 73
b < −4.4 or b > 3
Step 1: Isolate Absolute Value
2|10b + 7| > 74 →
Absolute Value Inequalities Practice Two INFINITE ALGEBRA 1 • LESSON 1-7 PRACTICE WORKSHEET 2 PAGE 1 OF 2
Absolute Value Inequalities
Solve Each Inequality and Graph Its Solution
Name:
Date:
Per:
Core Rules: |u| ≤ c → −c ≤ u ≤ c (AND / between) • |u| ≥ c → u ≥ c or u ≤ −c (OR / opposite rays) Problems 1–10
1 |n/4| ≤ 3
AND
Solution:
-12 -8 -4 0 4 8 12
2 |-9v| ≤ 54
AND
Solution:
-7 -6 0 6 8
3 |x/6| ≥ 5
OR
Solution:
-30 -20 -10 0 10 20 30
4 |-6b| ≤ 60
AND
Solution:
-12 -10 0 10 12
5 |-8n| < 32
AND
Solution:
-5 -4 0 4 7
6 |x + 5| < 9
AND
Solution:
-16 -14 0 4 6
7 |4v − 9| ≤ 27
AND
Solution:
-6 -4.5 0 9 10
8 |10 + 4x| < 14
AND
Solution:
-8 -6 0 1 4
Infinite Algebra 1 • Worksheet 2 • Lesson 1-7 Practice Turn page over for problems 9–22 Page 1 of 2
INFINITE ALGEBRA 1 • LESSON 1-7 PRACTICE WORKSHEET 2 PAGE 2 OF 2
Absolute Value Inequalities (Continued)
Two-Step & Multi-Step Operations
Name:
Isolate the absolute value completely before splitting into cases. Problems 9–22
9.|3 − 9a| ≤ 60 ≤
Sol:
-8 0 8
10.|7x + 4| ≥ 74 ≥
Sol:
-16 0 12
11.|n| − 3 > −2 >
Sol:
-6 0 6
12.|k| − 6 ≤ −1 ≤
Sol:
-8 0 7
13.|n| + 4 < 12 <
Sol:
-10 0 10
14.|x| + 7 > 16 >
Sol:
-12 0 12
15.|p| − 3 ≤ 0 ≤
Sol:
-6 0 6
16.|m| + 5 < 9 <
Sol:
-6 0 6
17.|b − 8| + 10 > 22 >
Sol:
-6 0 24
18.|(x − 4)/5| ≤ 2 ≤
Sol:
-6 0 16
19.−3 + |n − 2| > 5 >
Sol:
-8 0 12
20.|(3 + r)/7| ≤ 5 ≤
Sol:
Absolute Value Inequalities Key Two INFINITE ALGEBRA 1 • LESSON 1-7 TEACHER ANSWER KEY PAGE 1 OF 2
Absolute Value Inequalities
Worked Answer Key • Problems 1–10
Part 1: Basic & One-Step Inequalities
Red shaded lines and rays indicate solution sets. Solid dots = ≤/≥; open circles = </>. Problems 1–10
1 |n/4| ≤ 3
−12 ≤ n ≤ 12
−3 ≤ n/4 ≤ 3 → Multiply 4: −12 ≤ n ≤ 12
Interval: [−12, 12]
-12 0 12
2 |-9v| ≤ 54
−6 ≤ v ≤ 6
−54 ≤ −9v ≤ 54 → Divide −9 (reverse): −6 ≤ v ≤ 6
Interval: [−6, 6]
-7 -6 0 6 8
3 |x/6| ≥ 5
x ≤ −30 or x ≥ 30
x/6 ≥ 5 → x ≥ 30 or x/6 ≤ −5 → x ≤ −30
Interval: (−∞, −30] ∪ [30, ∞)
-30 0 30
4 |-6b| ≤ 60
−10 ≤ b ≤ 10
−60 ≤ −6b ≤ 60 → Divide −6: −10 ≤ b ≤ 10
Interval: [−10, 10]
-12 -10 0 10 12
5 |-8n| < 32
−4 < n < 4
−32 < −8n < 32 → Divide −8: −4 < n < 4
Interval: (−4, 4)
-5 -4 0 4 7
6 |x + 5| < 9
−14 < x < 4
−9 < x + 5 < 9 → Subtract 5: −14 < x < 4
Interval: (−14, 4)
-16 -14 0 4 6
7 |4v − 9| ≤ 27
−4.5 ≤ v ≤ 9
−27 ≤ 4v − 9 ≤ 27 → −18 ≤ 4v ≤ 36 → −9/2 ≤ v ≤ 9
Interval: [−4.5, 9]
-6 -4.5 0 9 10
8 |10 + 4x| < 14
−6 < x < 1
−14 < 10 + 4x < 14 → −24 < 4x < 4 → −6 < x < 1
Interval: (−6, 1)
-8 -6 0 1 4
Infinite Algebra 1 • Worksheet 2 Key Teacher Resource • Keep Confidential Page 1 of 2
INFINITE ALGEBRA 1 • LESSON 1-7 TEACHER ANSWER KEY PAGE 2 OF 2
Absolute Value Inequalities (Continued)
Worked Answer Key • Problems 9–22
Part 2: Multi-Step Inequalities
Red arrows indicate OR opposite rays. Shaded segments indicate AND bounded intervals. Problems 9–22
9.|3 − 9a| ≤ 60 −19/3 ≤ a ≤ 7
−60 ≤ 3 − 9a ≤ 60 → −63 ≤ −9a ≤ 57 → −19/3 ≤ a ≤ 7
-8 0 8
10.|7x + 4| ≥ 74 x ≤ −78/7 or x ≥ 10
7x + 4 ≥ 74 → x ≥ 10 or 7x + 4 ≤ −74 →
Exponents Review Guide Worksheet GRADE 8 MATHEMATICS • UNIT 1 REVIEW TEST REVIEW GUIDE PAGE 1 OF 2
Exponents & Scientific Notation
Comprehensive Test Review Guide
Name:
Date:
Instructions: Show all work. Simplify each expression or write in the requested form.
Part 1: Exponent Laws
A Multiplying Powers with the Same Base
Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
3⁴ × 3²
=
x⁵ × x³
=
7² × 7³
=
a⁴ × a
=
10³ × 10⁵
=
B Dividing Powers with the Same Base
Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
5⁶ ÷ 5²
=
x⁸ ÷ x³
=
9⁷ ÷ 9⁵
=
m⁴ ÷ m
=
12⁵ ÷ 12³
=
C Power of a Power
Rule: (aᵐ)ⁿ = aᵐ•ⁿ
(2³)⁴
=
(x²)⁵
=
(a³)³
=
(5⁴)²
=
(b⁶)³
=
D Power of a Product
Rule: (ab)ⁿ = aⁿbⁿ
(3x)²
=
(2ab)³
=
(4mn²)²
=
(5x²y)³
=
(6pq³)²
=
Grade 8 Mathematics • Exponent Properties Review Evaluate numerical expressions completely where appropriate Page 1 of 2
GRADE 8 MATHEMATICS • UNIT 1 REVIEW TEST REVIEW GUIDE PAGE 2 OF 2
Negative/Zero Powers & Scientific Notation
Part 2: Special Exponents & Scientific Notation
Name:
Instructions: Express negative exponents as fractions. Convert accurately between forms.
Problems 21–40
E Negative Exponents
Rule: a⁻ⁿ = 1 / aⁿ
Exponents Review Guide Answer Key GRADE 8 MATHEMATICS • UNIT 1 REVIEW TEACHER ANSWER KEY PAGE 1 OF 2
Exponents & Scientific Notation
Worked Answer Key • Problems 1–20
Focus: Exponent Product, Quotient & Power Laws
Teacher Note: Both the simplified exponential form and fully evaluated numerical values are shown below.
40 Questions Total
A Multiplying Powers with the Same Base
Rule: Add Exponents (m + n)
3⁴ × 3²
3⁴⁺² = 3⁶
3⁶ = 729
x⁵ × x³
x⁵⁺³
x⁸
7² × 7³
7²⁺³ = 7⁵
7⁵ = 16,807
a⁴ × a
a⁴ × a¹ = a⁴⁺¹
a⁵
10³ × 10⁵
10³⁺⁵ = 10⁸
10⁸
B Dividing Powers with the Same Base
Rule: Subtract Exponents (m − n)
5⁶ ÷ 5²
5⁶⁻² = 5⁴
5⁴ = 625
x⁸ ÷ x³
x⁸⁻³
x⁵
9⁷ ÷ 9⁵
9⁷⁻⁵ = 9²
9² = 81
m⁴ ÷ m
m⁴ ÷ m¹ = m⁴⁻¹
m³
12⁵ ÷ 12³
12⁵⁻³ = 12²
12² = 144
C Power of a Power
Rule: Multiply Exponents (m × n)
(2³)⁴
2³×⁴ = 2¹²
2¹² = 4,096
(x²)⁵
x²×⁵
x¹⁰
(a³)³
a³×³
a⁹
(5⁴)²
5⁴ײ = 5⁸
5⁸ = 390,625
(b⁶)³
b⁶׳
b¹⁸
D Power of a Product
Rule: Apply Exponent to Each Factor
(3x)²
3² × x²
9x²
(2ab)³
2³ × a³ × b³
8a³b³
(4mn²)²
4² × m² × (n²)²
16m²n⁴
(5x²y)³
5³ × (x²)³ × y³
Exponents Unit Practice Test GRADE 8 MATHEMATICS • UNIT 1 ASSESSMENT PRACTICE TEST PAGE 1 OF 2
Exponents & Scientific Notation
Unit Practice Test • Form A
Name:
Date: Per:
Score ____ / 50
Show all supporting steps clearly. Simplify completely. Part 1 & 2: 24 Points
Section 1: Exponent Laws (Problems 1–6 • 2 pts each) Product, Quotient, Power Rules
4³ × 4⁵
Answer:
x⁹ ÷ x⁴
Answer:
(3²)⁴
Answer:
(2a³b)⁴
Answer:
6x⁷y⁴ ÷ (2x³y)
Answer:
Equivalent to (5x⁴)³? [ ]
A) 15x⁷ B) 125x⁷ C) 125x¹² D) 15x¹²
Section 2: Zero & Negative Exponents (Problems 7–11) Express negative powers as positive fractions
12⁰ + (5y)⁰ (2 pts)
Value:
3⁻³ Evaluate fraction (2 pts)
Fraction:
5x⁻⁴ Positive exponent (2 pts)
Simplified:
(4m)⁻² Positive exponent (2 pts)
Simplified:
Error Analysis: Marcus simplified x³ ÷ x⁸ as x⁵. (4 pts) Constructed Response
A. Explain Marcus's Error:
B. Correct Simplified Form:
Correct Answer = ___________
Grade 8 Math • Exponents & Scientific Notation Practice Test Turn page over to continue with Part 3 & Part 4 Page 1 of 2
GRADE 8 MATHEMATICS • UNIT 1 ASSESSMENT PRACTICE TEST PAGE 2 OF 2
Scientific Notation & Real-World Modeling
Conversions, Operations & Astronomy Applications
Name:
Write numbers in standard form or proper scientific notation: a × 10ⁿ where 1 ≤ |a| < 10. Part 3 & 4: 26 Points
Section 3: Scientific Notation Conversions (12 pts • 2 pts each) Convert between standard form and scientific notation
Standard form of 4.8 × 10⁵
Standard form of 7.2 × 10⁻⁴
Standard form of 3.05 × 10³
Scientific notation for 670,000
Scientific notation for 0.000084
Scientific notation for 52,900,000
Section 4: Operations & Real-World Modeling (14 pts) Compare, calculate, and solve real applications
Exponents Practice Test Key GRADE 8 MATHEMATICS • UNIT 1 ASSESSMENT TEACHER ANSWER KEY PAGE 1 OF 2
Exponents & Scientific Notation
Practice Test Worked Key • Form A
Total: 50 Points Total (Key Included)
Award full credit for equivalent expressions unless a specific format was requested. Part 1 & 2: 24 Points
Section 1: Exponent Laws (Problems 1–6 • 2 pts each) 2 points each
4³ × 4⁵ 4³⁺⁵ = 4⁸
4⁸ = 65,536
x⁹ ÷ x⁴ x⁹⁻⁴
x⁵
(3²)⁴ 3²×⁴ = 3⁸
3⁸ = 6,561
(2a³b)⁴ 2⁴(a³)&sup4;b⁴
16a¹²b⁴
6x⁷y⁴ ÷ (2x³y) (6/2)x⁷⁻³y⁴⁻¹
3x⁴y³
(5x⁴)³ = 5³(x⁴)³ = 125x¹² C
A) 15x⁷ B) 125x⁷ C) 125x¹² ✓ D) 15x¹²
Section 2: Zero & Negative Exponents (Problems 7–11) 2 pts each (11 = 4 pts)
12⁰ + (5y)⁰ 1 + 1 (2 pts)
2
3⁻³ 1 / 3³ (2 pts)
1/27
5x⁻⁴ 5 × (1/x⁴) (2 pts)
5/x⁴
(4m)⁻² 1 / (4m)² (2 pts)
1/(16m²)
Error Analysis Key: Marcus simplified x³ ÷ x⁸ as x⁵. (4 pts) Rubric Included
A. Explanation (2 pts):
Marcus subtracted the exponents in reverse order (8 − 3 = 5) instead of top minus bottom: 3 − 8 = −5.
B. Correct Work (2 pts):
x³⁻⁸ = x⁻⁵ = 1 / x⁵
Answer: 1/x⁵ (or x⁻⁵)
Grade 8 Math • Practice Test Teacher Answer Key Teacher Resource • Keep Confidential Page 1 of 2
GRADE 8 MATHEMATICS • UNIT 1 ASSESSMENT TEACHER ANSWER KEY PAGE 2 OF 2
Scientific Notation & Real-World Modeling
Worked Answer Key • Problems 12–20
Section Total: 26 Points
Check for proper scientific notation form: 1 ≤ |a| < 10 with powers of 10. Part 3 & 4: 26 Points
Section 3: Scientific Notation Conversions (12 pts • 2 pts each) 2 points each
4.8 × 10⁵
480,000
7.2 × 10⁻⁴
0.00072
3.05 × 10³
3,050
670,000
6.7 × 10⁵
0.000084
8.4 × 10⁻⁵
Scientific Notation Study Guide GRADE 8 MATHEMATICS • LESSON 9 QUIZ STUDY GUIDE PAGE 1 OF 2
Scientific Notation Study Guide
Lesson 9: Conversions & Real Numbers
Name:
Date:
Per:
1
Part 1 Rule: Scientific Notation → Real (Standard) Form Move the decimal point the number of places indicated by the exponent n :
10⁺ⁿ: Move RIGHT
10⁻ⁿ: Move LEFT
1 5.834 × 10⁻¹
1 place left
Standard Form:
2 6.952 × 10⁻³
3 places left
Standard Form:
3 6.418 × 10²
2 places right
Standard Form:
4 5.131 × 10⁻³
3 places left
Standard Form:
5 8.997 × 10⁻⁴
4 places left
Standard Form:
6 3.431 × 10⁻⁶
6 places left
Standard Form:
7 7.317 × 10³
3 places right
Standard Form:
8 5.897 × 10⁻¹
1 place left
Standard Form:
9 8.517 × 10¹
1 place right
Standard Form:
10 7.464 × 10⁻⁵
5 places left
Standard Form:
Lesson 9: Scientific Notation Study Guide • WorksheetWorks Review Turn page over for Real → Scientific Notation conversions Page 1 of 2
GRADE 8 MATHEMATICS • LESSON 9 QUIZ STUDY GUIDE PAGE 2 OF 2
Converting Real to Scientific Notation
Form: a × 10ⁿ where 1 ≤ |a| < 10
Name:
2
Part 2 Rule: Real (Standard) Number → Scientific Notation Place the decimal after the first non-zero digit. Count the number of places moved:
Moved left → 10⁺ⁿ
Moved right → 10⁻ⁿ
No move → 10⁰
11 42.62
Decimal moves left
Sci Notation:
12 8,843,000
Decimal moves left
Sci Notation:
13 938,600
Decimal moves left
Sci Notation:
14 34,150
Decimal moves left
Sci Notation:
15 42.13
Decimal moves left
Sci Notation:
16 0.000006967
Decimal moves right
Sci Notation:
Scientific Notation Study Guide Key GRADE 8 MATHEMATICS • LESSON 9 TEACHER ANSWER KEY PAGE 1 OF 2
Scientific Notation Study Guide
Worked Answer Key • Problems 1–10
Part 1: Scientific → Standard Form
Part 1 Key Notes Shift decimal right for positive powers (10⁺); shift left for negative powers (10⁻).
10 Questions
1 5.834 × 10⁻¹
1 place left
0.5834 (decimal shifts 1 left)
Standard: 0.5834
2 6.952 × 10⁻³
3 places left
0.006952 (add 2 leading zeros)
Standard: 0.006952
3 6.418 × 10²
2 places right
641.8 (decimal shifts 2 right)
Standard: 641.8
4 5.131 × 10⁻³
3 places left
0.005131 (add 2 leading zeros)
Standard: 0.005131
5 8.997 × 10⁻⁴
4 places left
0.0008997 (add 3 leading zeros)
Standard: 0.0008997
6 3.431 × 10⁻⁶
6 places left
0.000003431 (add 5 leading zeros)
Standard: 0.000003431
7 7.317 × 10³
3 places right
7,317 (decimal shifts 3 right)
Standard: 7,317
8 5.897 × 10⁻¹
1 place left
0.5897 (decimal shifts 1 left)
Standard: 0.5897
9 8.517 × 10¹
1 place right
85.17 (decimal shifts 1 right)
Standard: 85.17
10 7.464 × 10⁻⁵
5 places left
0.00007464 (add 4 leading zeros)
Standard: 0.00007464
Lesson 9 Study Guide Key • Teacher Resource Confidential Key • Page 1 of 2 Page 1 of 2
GRADE 8 MATHEMATICS • LESSON 9 TEACHER ANSWER KEY PAGE 2 OF 2
Converting Real to Scientific Notation
Worked Answer Key • Problems 11–22
Part 2: Standard → Scientific Notation
Part 2 Key Notes Count places to reach 1 ≤ |a| < 10. Note: Problem 21 is already between 1 and 10 → power is 10⁰!
12 Questions
11 42.62
1 place left
42.62 → 4.262 × 10¹
Sci Form: 4.262 × 10¹
12 8,843,000
6 places left
8,843,000. → 8.843 × 10⁶
Sci Form: 8.843 × 10⁶
13 938,600
5 places left
938,600. → 9.386 × 10⁵
Sci Form: 9.386 × 10⁵
14 34,150
