One Step Guided Notes 8th Grade Math • Guided Notes
One-Step Equations & Inequalities
Name:
Date:
Period:
Core Concept 1: The Golden Rule of Algebra
An equation represents a balanced scale . To keep both sides equal, whatever operation you perform on one side, you must do to the side.
To isolate the variable, use inverse operations (operations that each other).
Inverse Pairs
Addition \(+\) \(\longleftrightarrow\) Subtraction \(-\)
Multiply \(\times\) \(\longleftrightarrow\) Divide \(\div\)
Core Concept 2: The Inequality Flip Rule
Must Memorize
When you multiply or divide both sides of an inequality by a number, you MUST (reverse) the inequality sign!
\(>\) flips to \(<\) • \(<\) flips to \(>\) • \(\ge\) flips to \(\le\) • \(\le\) flips to \(\ge\)
Graphing on a Number Line
\(<\) or \(>\) → Open circle \(\circ\)
\(\le\) or \(\ge\) → Closed circle \(\bullet\)
\(\rightarrow\) Greater (Right) Less (Left) \(\leftarrow\)
Guided Examples • Step-by-Step Modeling
Example 1 Addition Equation
\( x + 7 = -4 \)
1. Identify the operation applied to \(x\):
2. Apply the inverse: from both sides .
Check: \(( \quad ) + 7 = -4\)
Solution:
Example 2 Division Equation
\( \frac{m}{-3} = 6 \)
1. Identify the operation applied to \(m\):
2. Apply the inverse: Multiply both sides by .
Check: \(\frac{( \quad )}{-3} = 6\)
Solution:
Example 3 No Flip Rule Needed
\( 4y \ge -12 \)
Divide by +4. Is the divisor negative? [ YES / NO ] [ YES / NO ]
Solution:
-5 -4 -3 -2 -1 0 1
Example 4 Flip Rule Applies!
\( -2w < 8 \)
Dividing by -2 (negative!).\(<\) to !
Solution:
-7 -6 -5 -4 -3 -2 -1
Unit 2: Linear Foundations • Guided Notes One-Step Equations & Inequalities
One Step Notes Answer Key 8th Grade Math • Guided Notes Teacher Answer Key
One-Step Equations & Inequalities
Name: Answer Key
Date: Complete
Period: All
Core Concept 1: The Golden Rule of Algebra
An equation represents a balanced scale . To keep both sides equal, whatever operation you perform on one side, you must do to the other side.
To isolate the variable, use inverse operations (operations that undo / cancel each other).
Inverse Pairs
Addition \(+\) \(\longleftrightarrow\) Subtraction \(-\)
Multiply \(\times\) \(\longleftrightarrow\) Divide \(\div\)
Core Concept 2: The Inequality Flip Rule
Teacher Note: Emphasize Negative Divisor
When you multiply or divide both sides of an inequality by a negative number, you MUST flip / reverse (reverse) the inequality sign!
\(>\) flips to \(<\) • \(<\) flips to \(>\) • \(\ge\) flips to \(\le\) • \(\le\) flips to \(\ge\)
Graphing on a Number Line
\(<\) or \(>\) → Open circle \(\circ\)
\(\le\) or \(\ge\) → Closed circle \(\bullet\)
\(\rightarrow\) Greater (Right) Less (Left) \(\leftarrow\)
Worked Model Examples • Answer Key
Example 1 Addition Equation
\( x + 7 = -4 \)
1. Identify the operation applied to \(x\): Addition (+7)
2. Apply the inverse: Subtract 7 from both sides .
\(x + 7 = -4\) Subtract 7
\(\phantom{x} - 7 \quad -7\) Balance property
\(x = -11\)
Check: \((\mathbf{-11}) + 7 = -4 \;\checkmark\)
Solution:
\(x = -11\)
Example 2 Division Equation
\( \frac{m}{-3} = 6 \)
1. Identify the operation applied to \(m\): Division (\(\div -3\))
2. Apply the inverse: Multiply both sides by -3.
\(\frac{m}{-3} = 6\) Multiply by \(-3\)
\((-3)\cdot\frac{m}{-3} = 6\cdot(-3)\) Cancel denominator
\(m = -18\)
Check: \(\frac{\mathbf{-18}}{-3} = 6 \;\checkmark\)
Solution:
\(m = -18\)
Example 3 No Flip Rule Needed
\( 4y \ge -12 \)
Divide by +4. Is the divisor negative? [ NO ] [ NO ] • Keep \(\ge\)
\(\frac{4y}{4} \ge \frac{-12}{4}\) Divisor \(+4 > 0\)
\(y \ge -3\)
Solution: \(y \ge -3\) Closed circle at -3, arrow RIGHT
-5 -4 -3 -2 -1 0 1
Example 4 Flip Rule Applies!
\( -2w < 8 \)
Dividing by -2 (negative!).\(<\) to \(>\) !
\(\frac{-2w}{-2} \mathbf{>} \frac{8}{-2}\) FLIP \(<\) to \(>\)
\(w > -4\)
Solution: \(w > -4\) Open circle at -4, arrow RIGHT
-7 -6 -5 -4 -3 -2 -1
Unit 2: Linear Foundations • Guided Notes Key One-Step Equations & Inequalities
One Step Equations Homework 8th Grade Math • Linear Foundations
One-Step Equations
Name:
Date:
Period:
Directions: Solve each equation below. Show your inverse operations clearly in the space provided. Write your final answer in simplest form (as an integer or simplified fraction).
1 Addition Equation
\( x + 9 = 4 \)
Solution:
2 Subtraction Equation
\( m - 14 = -6 \)
Solution:
3 Multiplication Equation
\( -7y = 56 \)
Solution:
4 Division Equation
\( \frac{w}{5} = -9 \)
Solution:
5 Multiplication Equation
\( 4k = 18 \)
Solution:
6 Subtraction Equation
\( p - 8 = -23 \)
Solution:
Unit 2: Linear Equations • Homework 1 Page 1 of 2
8th Grade Math • Linear Foundations
One-Step Inequalities
Name:
Date:
Period:
Directions: Solve each inequality showing all inverse operations. Remember: reverse the inequality symbol when multiplying or dividing by a negative. Then graph each solution on the number line.
7 Solve & Graph
\( x + 6 \le 2 \)
Solution:
-7 -6 -5 -4 -3 -2 -1
8 Solve & Graph
\( y - 5 > -2 \)
Solution:
0 1 2 3 4 5 6
9 Solve & Graph
\( -3m \ge 15 \)
Solution:
-8 -7 -6 -5 -4 -3 -2
10 Solve & Graph
\( \frac{a}{4} < -2 \)
Solution:
-11 -10 -9 -8 -7 -6 -5
11 Solve & Graph
\( 6b > 9 \)
Solution:
0 ½ 1 1½ 2 2½ 3
12 Solve & Graph
\( -5n < -30 \)
Solution:
3 4 5 6 7 8 9
Unit 2: Linear Equations • Homework 1 Page 2 of 2
One Step Solutions Answer Key 8th Grade Math • Teacher Solutions Answer Key
One-Step Equations — Key
Homework 1 • Page 1
Full Working Steps Included
Scoring Guidance: Award 1 point for applying the correct inverse operation to both sides and 1 point for the correct simplified solution. Problem 5 results in an improper fraction in simplest form or mixed number.
1 Subtract 9 from both sides
\( x + 9 = 4 \)
\( x + 9 - 9 = 4 - 9 \)
\( x + 0 = -5 \)
Solution:
\( x = -5 \)
2 Add 14 to both sides
\( m - 14 = -6 \)
\( m - 14 + 14 = -6 + 14 \)
\( m + 0 = 8 \)
Solution:
\( m = 8 \)
3 Divide both sides by -7
\( -7y = 56 \)
\( \frac{-7y}{-7} = \frac{56}{-7} \)
\( 1y = -8 \)
Solution:
\( y = -8 \)
4 Multiply both sides by 5
\( \frac{w}{5} = -9 \)
\( 5 \cdot \left(\frac{w}{5}\right) = 5 \cdot (-9) \)
\( w = -45 \)
Solution:
\( w = -45 \)
5 Divide by 4 • Simplify fraction
\( 4k = 18 \)
\( k = \frac{18}{4} = \frac{18 \div 2}{4 \div 2} \)
\( k = \frac{9}{2} \text{ or } 4\frac{1}{2} \text{ or } 4.5 \)
Solution:
\( k = \frac{9}{2} \)
6 Add 8 to both sides
\( p - 8 = -23 \)
\( p - 8 + 8 = -23 + 8 \)
\( p = -15 \)
Solution:
\( p = -15 \)
Unit 2: Linear Equations • Homework 1 Answer Key Page 1 of 2
8th Grade Math • Teacher Solutions Answer Key
One-Step Inequalities — Key
Homework 1 • Page 2
Graphed Number Lines Included
Graphing Rules: Use a solid circle for \(\le\) and \(\ge\); use an open circle for \(<\) and \(>\). Division/multiplication by negative numbers reverses inequality direction (#9 and #12).
7 Subtract 6 • Closed circle at -4
\( x + 6 \le 2 \implies x \le -4 \)
\( x + 6 - 6 \le 2 - 6 \implies x \le -4 \)
Solution:
\( x \le -4 \)
-7 -6 -5 -4 -3 -2 -1
8 Add 5 • Open circle at 3
\( y - 5 > -2 \implies y > 3 \)
\( y - 5 + 5 > -2 + 5 \implies y > 3 \)
Solution:
\( y > 3 \)
0 1 2 3 4 5 6
9 Divide by -3 • FLIP sign to ≤
Two Step Guided Notes 8th Grade Math • Guided Notes
Two-Step Equations & Inequalities
Name:
Date:
Period:
Core Roadmap: Two-Step Solving Process
1 Undo Addition or Subtraction
Use inverse operations to isolate the term first (get constants onto the other side).
2 Undo Multiplication or Division
Use inverse operations to isolate the completely (get a coefficient of \(1\)).
Inequality Check: When Do You Flip The Symbol?
Watch Out!
In Step 2 , ONLY flip the symbol if you multiply or divide by a negative number !
A negative constant on the right side does NOT flip the symbol!
FLIP THE SYMBOL!
\( -3x > 12 \;\xrightarrow{\div -3}\; x < -4 \)
Dividing by negative 3
DO NOT FLIP!
\( 3x > -12 \;\xrightarrow{\div 3}\; x > -4 \)
Dividing by positive 3
Guided Examples • Step-by-Step Modeling
Example 1 Multiply & Add
\( 3x + 7 = 22 \)
Step 1: 7 from both sides.
Step 2: Divide both sides by .
Check: \(3( \quad ) + 7 = 22\)
Solution:
Example 2 Divide & Subtract
\( \frac{w}{3} - 4 = 2 \)
Step 1: Add to both sides.
Step 2: Multiply both sides by .
Check: \(\frac{( \quad )}{3} - 4 = 2\)
Solution:
Example 3 No Flip Rule Needed
\( 2x + 5 \le 17 \)
Step 1: Subtract 5 • Step 2: Divide by +2 .[ NO ] • Flip symbol? [ NO ]
Solution:
3 4 5 6 7 8 9
Example 4 Flip Rule Applies!
\( -3m + 8 \ge 29 \)
Step 1: Subtract 8 • Step 2: Divide by -3.
Solution:
-10 -9 -8 -7 -6 -5 -4
Unit 2: Linear Foundations • Guided Notes Two-Step Equations & Inequalities
Two Step Notes Answer Key 8th Grade Math • Guided Notes Teacher Answer Key
Two-Step Equations & Inequalities
Name: Answer Key
Date: Complete
Period: All
Core Roadmap: Two-Step Solving Process
1 Undo Addition or Subtraction
Use inverse operations to isolate the variable term first (get constants onto the other side).
2 Undo Multiplication or Division
Use inverse operations to isolate the variable completely (get a coefficient of \(1\)).
Inequality Check: When Do You Flip The Symbol?
Critical Distinction
In Step 2 , ONLY flip the symbol if you multiply or divide by a negative number !
A negative constant on the right side does NOT flip the symbol!
FLIP THE SYMBOL!
\( -3x > 12 \;\xrightarrow{\div -3}\; x < -4 \)
Dividing by negative 3
DO NOT FLIP!
\( 3x > -12 \;\xrightarrow{\div 3}\; x > -4 \)
Dividing by positive 3
Worked Model Examples • Answer Key
Example 1 Multiply & Add
\( 3x + 7 = 22 \)
Step 1: Subtract 7 from both sides.
Step 2: Divide both sides by 3.
\(3x + 7 = 22\) Subtract 7
\(\phantom{3x} - 7 \quad -7\) \(3x = 15\)
\(\frac{3x}{3} = \frac{15}{3}\) \(x = 5\)
Check: \(3(\mathbf{5}) + 7 = 15 + 7 = 22 \;\checkmark\)
Solution:
\(x = 5\)
Example 2 Divide & Subtract
\( \frac{w}{3} - 4 = 2 \)
Step 1: Add 4 to both sides.
Step 2: Multiply both sides by 3.
\(\frac{w}{3} - 4 = 2\) Add 4
\(\phantom{\frac{w}{3}} + 4 \quad +4\) \(\frac{w}{3} = 6\)
\(3 \cdot \left(\frac{w}{3}\right) = 6 \cdot 3\) \(w = 18\)
Check: \(\frac{\mathbf{18}}{3} - 4 = 6 - 4 = 2 \;\checkmark\)
Solution:
\(w = 18\)
Example 3 No Flip Rule Needed
\( 2x + 5 \le 17 \)
Step 1: Subtract 5 • Step 2: Divide by +2 .[ NO ] • Flip symbol? [ NO ]
\(2x + 5 \le 17 \;\implies\; 2x \le 12\) Subtract 5
\(\frac{2x}{2} \le \frac{12}{2}\) \(x \le 6\)
Solution: \(x \le 6\) Closed circle at 6, arrow LEFT
3 4 5 6 7 8 9
Example 4 Flip Rule Applies!
\( -3m + 8 \ge 29 \)
Two Step Equations Homework 8th Grade Math • Linear Foundations
Two-Step Equations
Name:
Date:
Period:
Directions: Solve each two-step equation. First isolate the variable term using addition or subtraction, then isolate the variable using multiplication or division. Show all steps clearly.
1 Two-Step Equation
\( 3x + 7 = 22 \)
Solution:
2 Two-Step Equation
\( 5y - 12 = -37 \)
Solution:
3 Two-Step Equation
\( -4m + 9 = -19 \)
Solution:
4 Two-Step Equation
\( \frac{w}{3} - 4 = 2 \)
Solution:
5 Two-Step Equation
\( 6k + 4 = 19 \)
Solution:
6 Two-Step Equation
\( -\frac{p}{2} + 11 = 3 \)
Solution:
Unit 2: Linear Equations • Homework 2 Page 1 of 2
8th Grade Math • Linear Foundations
Two-Step Inequalities
Name:
Date:
Period:
Directions: Solve each two-step inequality showing all inverse operations. Remember: flip the inequality sign when multiplying or dividing by a negative number. Graph your final solution on the number line.
7 Solve & Graph
\( 2x + 5 \le 17 \)
Solution:
3 4 5 6 7 8 9
8 Solve & Graph
\( 4y - 9 > 7 \)
Solution:
1 2 3 4 5 6 7
9 Solve & Graph
\( -3m + 8 \ge 29 \)
Solution:
-10 -9 -8 -7 -6 -5 -4
10 Solve & Graph
\( \frac{w}{5} + 3 < 1 \)
Solution:
-13 -12 -11 -10 -9 -8 -7
11 Solve & Graph
\( 4k - 3 \le 3 \)
Solution:
0 ½ 1 1½ 2 2½ 3
12 Solve & Graph
\( -2p - 7 < 5 \)
Solution:
-9 -8 -7 -6 -5 -4 -3
Unit 2: Linear Equations • Homework 2 Page 2 of 2
Two Step Solutions Answer Key 8th Grade Math • Teacher Solutions Answer Key
Two-Step Equations — Key
Homework 2 • Page 1
Full Working Steps Included
Scoring Guidance: Award 1 point for the first inverse operation (undoing addition/subtraction), 1 point for the second inverse operation (undoing multiplication/division), and 1 point for the correct simplified solution.
1 Subtract 7, then divide by 3
\( 3x + 7 = 22 \)
\( 3x + 7 - 7 = 22 - 7 \implies 3x = 15 \)
\( \frac{3x}{3} = \frac{15}{3} \implies x = 5 \)
Solution:
\( x = 5 \)
2 Add 12, then divide by 5
\( 5y - 12 = -37 \)
\( 5y - 12 + 12 = -37 + 12 \implies 5y = -25 \)
\( \frac{5y}{5} = \frac{-25}{5} \implies y = -5 \)
Solution:
\( y = -5 \)
3 Subtract 9, then divide by -4
\( -4m + 9 = -19 \)
\( -4m + 9 - 9 = -19 - 9 \implies -4m = -28 \)
\( \frac{-4m}{-4} = \frac{-28}{-4} \implies m = 7 \)
Solution:
\( m = 7 \)
4 Add 4, then multiply by 3
\( \frac{w}{3} - 4 = 2 \)
\( \frac{w}{3} - 4 + 4 = 2 + 4 \implies \frac{w}{3} = 6 \)
\( 3 \cdot \left(\frac{w}{3}\right) = 3 \cdot 6 \implies w = 18 \)
Solution:
\( w = 18 \)
5 Subtract 4, divide by 6 • Fraction
\( 6k + 4 = 19 \)
\( 6k = 19 - 4 \implies 6k = 15 \)
\( k = \frac{15}{6} = \frac{5}{2} \text{ or } 2\frac{1}{2} \text{ or } 2.5 \)
Solution:
\( k = \frac{5}{2} \)
6 Subtract 11, multiply by -2
\( -\frac{p}{2} + 11 = 3 \)
\( -\frac{p}{2} = 3 - 11 \implies -\frac{p}{2} = -8 \)
\( (-2) \cdot \left(-\frac{p}{2}\right) = (-2) \cdot (-8) \implies p = 16 \)
Solution:
\( p = 16 \)
Unit 2: Linear Equations • Homework 2 Answer Key Page 1 of 2
8th Grade Math • Teacher Solutions Answer Key
Two-Step Inequalities — Key
Homework 2 • Page 2
Graphed Number Lines Included
Graphing Rules: Solid circles for \(\le\) and \(\ge\); open circles for \(<\) and \(>\). Notice sign reversals on problems #9 (divided by -3) and #12 (divided by -2).
7 Subtract 5, divide 2 • Solid dot at 6
\( 2x + 5 \le 17 \implies 2x \le 12 \implies x \le 6 \)
\( 2x \le 17 - 5 \implies 2x \le 12 \implies x \le 6 \)
Solution:
\( x \le 6 \)
