ACT Math Lecture Notes

Introduction

• Objective: Solve common ACT math questions.
• Method: Pause video, attempt solutions, and check answers.

Problem 1: Solving Equations

• Equation: ( \frac{3x}{4} + 5 = 11 )
  • Steps:
    1. Subtract 5 from both sides: ( 11 - 5 = 6 )
    2. Multiply both sides by 4: ( 6 \times 4 = 24 )
    3. Divide by 3: ( 24 \div 3 = 8 )
• Expression: ( \frac{5x}{2} - 16 )
  • Substitute ( x = 8 )
  • ( \frac{5 imes 8}{2} - 16 = 4 )
• Answer: D

Problem 2: Sum of Solutions

• Equation: ( x^2 + 4x - 45 = 0 )
  • Factor: (x - 5)(x + 9)
  • Solutions: ( x = 5 ) and ( x = -9 )
• Sum: ( 5 + (-9) = -4 )
• Answer: B

Problem 3: Function Evaluation

• Function: ( f(x) = 5x^3 - 2x + 7 )
  • ( f(2) = 43 )
• Answer: E

Problem 4: Area and Perimeter of a Square

• Square Area: 36
  • Side Length: ( \sqrt{36} = 6 )
• Perimeter: 4 sides ( \times 6 = 24 )
• Answer: C

Problem 5: Quadratic Equation

• Equation: ( 3x^2 - 5x + 8 = 36 )
  • Subtract 36: ( 3x^2 - 5x - 28 = 0 )
• Factoring Method:
  • Multiply to (-84), factor to (-12) and (7)
  • Solutions: ( x = 4 ) and ( x = -\frac{7}{3} )
• Quadratic Formula: Alternative method confirming ( x = 4 )
• Answer: D

Problem 6: Equation from Graph

• Form: Slope-Intercept (( y = mx + b ))
• Y-Intercept: -3
• Slope (Rise/Run): ( \frac{3}{4} )
• Answer: B

Problem 7: Triangle Area

• Angle: 45 degrees
• Formula: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} )
• Find sides using Pythagorean Theorem
• Area: 25 square units
• Answer: B

Problem 8: Average Test Score

• Scores: 86, 93, 79, 82, 90
• Average: ( \frac{430}{5} = 86 )
• Answer: C

Problem 9: Average with New Score

• New Score: 99
• New Average: 87
• Steps: Add scores and divide by 7
• Answer: D

Problem 10: Area of Shaded Region

• Equation: ( \text{Circle Area} - \text{Triangle Area} )
• Steps: Calculate individual areas
• Answer: C

Problem 11: Trigonometric Ratios

• Given: ( \sin(z) = -\frac{7}{25} ), ( \tan(z) < 0 )
• Quadrant: 4
• Calculate: ( \cos(z) = \frac{24}{25} )
• Answer: D

Problem 12: Sum and Reciprocal

• Equation: ( x + \frac{6}{x} = 5 )
• Solve for x: Values are 2 and 3
• Expression: 5 + 4x
• Correct x: 2 yields 13
• Answer: D

Problem 13: System of Equations

• Equations:
  1. ( 3x - 4y = 3 )
  2. ( 2x + 3y = 19 )
• Solve: ( x = 5, y = 3 )
• Expression: ( x^2 - y^2 = 16 )
• Answer: C

Problem 14: Cost Calculations

• Equations for costs:
  1. ( 7a + 8b = 13.95 )
  2. ( 9a + 6b = 15.15 )
• Objective: Find cost for 11 apples, 12 bananas
• Answer: D

Problem 15: Exponent Properties

• Expression: ( y^4^5 \times y^{-8} )
• Simplified: ( \frac{1}{y^4} )
• Answer: D

Problem 16: Slope Calculation

• Equation: Standard to Slope-Intercept
• Calculation: Slope = ( \frac{3}{5} )
• Answer: C

Problem 17: Integer Inequality

• Inequality: Solve for R values
• Sum: Largest and smallest
• Answer: D

Problem 18: Circle Equation

• Formula: ((x-h)^2 + (y-k)^2 = r^2)
• Center: (2,1)
• Radius: 3
• Answer: A

Problem 19: Trigonometric Simplification

• Expression: Simplify using identities
• Answer: B

Problem 20: Trigonometric Expression

• Method: Multiply by conjugate
• Simplify: Result is cosine
• Answer: B

Problem 21: Triangle Side Calculation

• Area & Base Calculation
• Result: Side length 8
• Answer: C

Problem 22: Sales Tax Calculation

• Original Price: 70
• Tax: 8%
• Final Price: 75.60
• Answer: C

Problem 23: Reverse Sales Tax

• Total Price: 131.43
• Original without Tax: 123.99
• Answer: D

Problem 24: Discount Calculations

• Final Price: 196 after 20% discount
• Original Price: 245
• Answer: D

Problem 25: Two-Step Price Calculation

• Original Price: 300
• Discount & Tax: Calculate final cost
• Change Given: 7.15
• Answer: D

Problem 26: Cylinder Volume

• Volume: 80 pi
• Radius: 4, find height
• Height: 5
• Answer: B

Problem 27: Slope Between Points

• Points: (1/2, -2/3) and (-2/5, 1/4)
• Slope Calculation: (-\frac{55}{54})
• Answer: E

Problem 28: Proportion Problem

• Apples per time: Calculate using proportion
• Result: 10.5 apples
• Answer: A

Problem 29: Distance and Time

• Travel Proportion: Solve for hours
• Result: (\frac{14}{3}) hours
• Answer: D

Problem 30: Multi-step Muffin Problem

• Conversion Factors: Solve using stepwise conversions
• Result: 120 cups of flour
• Answer: D