Algebra 1 Lecture Notes

Introduction

• Focus on solving Algebra 1 questions easily.
• More detailed content available at ultimatealgebra.com.

Solving Equations

One-Step Equations

Example 1: Solve x + 2 = 5

• Goal: Isolate x on one side.
• Opposite operation of addition is subtraction.
• Steps:
  • Subtract 2 from both sides: 5 - 2 = 3
  • Result: x = 3

Two-Step Equations

Example 2: Solve 2x + 3 = 11

• Goal: Isolate x using reversal of the order of operations (PEMDAS).
• Steps:
  • Subtract 3 from both sides: 11 - 3 = 8
  • Divide both sides by 2: 8 / 2 = 4
  • Result: x = 4

Multi-Step Equations

Example 3: Solve 3x^2 + 8 = 20

• Goal: Isolate x by using order of operations in reverse.
• Steps:
  • Subtract 8 from both sides: 20 - 8 = 12
  • Divide both sides by 3: 12 / 3 = 4
  • Take the square root of both sides: √4 = 2
  • Result: x = 2

Complex Equations

Equation with Variable on Both Sides

Example 4: Solve 4x + 5 = 9 + 2x

• Goal: Move all x terms to one side.
• Steps:
  • Subtract 2x from both sides: 4x - 2x = 2x
  • Subtract 5 from both sides: 9 - 5 = 4
  • Divide both sides by 2: 4 / 2 = 2
  • Result: x = 2

Absolute Value Equations

Example 5: Solve |x + 3| = 7

• Goal: Consider positive and negative scenarios.
• Steps:
  • x + 3 = 7 -> x = 4
  • x + 3 = -7 -> x = -10
  • Result: x = 4 or x = -10

Example 6: Solve |x + 1| + 6 = 9

• Goal: Isolate absolute value term.
• Steps:
  • Subtract 6 from both sides: 9 - 6 = 3
  • x + 1 = 3 -> x = 2
  • x + 1 = -3 -> x = -4
  • Result: x = 2 or x = -4

Radical Equations

Example 7: Solve √(x + 3) - 2 = 1

• Goal: Isolate the radical.
• Steps:
  • Add 2 to both sides: 1 + 2 = 3
  • Square both sides: 3^2 = 9
  • Subtract 3 from both sides: 9 - 3 = 6
  • Result: x = 6

Rational Equations

Example 8: Solve 4 / (x - 5) = 3 / x

• Goal: Cross-multiply to clear fractions.
• Steps:
  • 4x = 3(x - 5) -> 4x = 3x - 15
  • Subtract 3x from both sides: x = -15
  • Result: x = -15

Changing the Subject of a Formula

Example 9: Solve y = mx + b for x

• Goal: Isolate x.
• Steps:
  • Subtract b from both sides: y - b = mx
  • Divide both sides by m: (y - b) / m = x
  • Result: x = (y - b) / m

Inequalities

Solving Inequalities

Example 10: Solve -3x + 1 > 7

• Goal: Isolate x.
• Steps:
  • Subtract 1 from both sides: 7 - 1 = 6
  • Divide both sides by -3 -> flip inequality: 6 / -3 = -2
  • Result: x < -2

Combined Inequalities

Example 11: Solve -3 < x + 8 < 20

• Goal: Isolate x.
• Steps:
  • Subtract 8 from all sides: -11 < x < 12
  • Result: -11 < x < 12

Graphing Inequalities

Example 12: Graph x > -4

• Goal: Represent inequality on a number line.
• Steps:
  • Locate the point -4.
  • Draw an unshaded circle at -4.
  • Draw arrow to the right.

Word Problems

Example 13: Shipping Problem

• Total gallons = 2500, boxes = 20, leftover = 100
• Equation: 20x + 100 = 2500
• Steps:
  • Subtract 100 from both sides: 2500 - 100 = 2400
  • Divide by 20: 2400 / 20 = 120
  • Result: 120 gallons per box

Example 14: Age Problem

• Equation: 5 + 3x = 50
• Steps:
  • Subtract 5 from both sides: 50 - 5 = 45
  • Divide by 3: 45 / 3 = 15
  • Result: Michael is 15 years old

Identifying Functions

Example 15: Determine if Relations are Functions

• Function: No input value can have more than one output value.
• Example:
  • (2 → 7), (2 → 5) is not a function.
  • (4 → 8), (5 → 8) is a function.
  • Answer: C (3 → 6, 3 → 8) is not a function.