Ultimate Algebra Lecture Notes: Solving Algebra 1 Questions

Welcome to Ultimate Algebra! This lecture covers various types of Algebra 1 problems and the easiest methods to solve them. For a more comprehensive course, visit our website at ultimatealgebra.com.

One-Step Equations

Example 1: Solve x + 2 = 5

• Objective: Isolate x.
• Steps:
  1. Subtract 2 from both sides.
  2. The equation simplifies to x = 3.

Two-Step Equations

Example 2: Solve 2x + 3 = 11

• Objective: Isolate x.
• Steps:
  1. Subtract 3 from both sides: 2x = 8.
  2. Divide by 2: x = 4.

Multi-Step Equations

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

• Objective: Isolate x.
• Steps:
  1. Subtract 8 from both sides.
  2. Divide by 3.
  3. Take the square root.
  4. Result: x = 2.

Equations with Variables on Both Sides

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

• Objective: Isolate x.
• Steps:
  1. Subtract 2x from both sides: 2x + 5 = 9.
  2. Subtract 5 from both sides: 2x = 4.
  3. Divide by 2: x = 2.

Absolute Value Equations

Example 5: Solve |x + 3| = 7

• Objective: Solve for x.
• Steps:
  1. Set up two equations: x + 3 = 7 and x + 3 = -7.
  2. Solve both: x = 4 and x = -10.

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

• Objective: Isolate the absolute value term.
• Steps:
  1. Subtract 6 from both sides: |x + 1| = 3.
  2. Set up two equations: x + 1 = 3 and x + 1 = -3.
  3. Solve both: x = 2 and x = -4.

Radical Equations

Example 7: Solve √(x + 3) - 2 = 1

• Objective: Isolate the radical.
• Steps:
  1. Add 2 to both sides: √(x + 3) = 3.
  2. Square both sides: x + 3 = 9.
  3. Subtract 3: x = 6.

Rational Equations

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

• Objective: Remove the fractions.
• Steps:
  1. Cross multiply: 4x = 3(x - 5).
  2. Simplify and solve: x = -5.

Change of Subject (Transposing Formulas)

Example 9: Solve for x in y = mx + b

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

Inequalities

Example 10: Solve -3x + 1 > 7

• Objective: Isolate x and note the change of the inequality sign when dividing by a negative number.
• Steps:
  1. Subtract 1 from both sides: -3x > 6.
  2. Divide by -3 (and flip inequality sign): x < -2.

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

• Objective: Isolate x in a combined inequality.
• Steps:
  1. Subtract 8 from all parts: -11 < x < 12.

Graphing Inequalities

Example 12: Graph x > -4

• Steps:
  1. Locate -4 on the number line.
  2. Draw an unshaded circle at -4.
  3. Draw an arrow to the right to indicate x > -4.

Word Problems

Example 13: Packaging Gallons

• Problem: Ship 2500 gallons with 20 boxes and 100 gallons left over. Find gallons per box.
• Steps:
  1. Setup equation: 20x + 100 = 2500
  2. Subtract 100 then divide by 20: x = 120.

Example 14: Michael's Age

• Problem: 5 + 3 times Michael's age = 50. Find Michael's age.
• Steps:
  1. Setup equation: 5 + 3x = 50
  2. Subtract 5 then divide by 3: x = 15.

Functions vs Relations

Example 15: Identify non-functions

• Key points:
  • No input value (x) should have multiple output values (y).
  • Every input value must have an output value.
• Example that is not a function: An input 3 going to outputs 6 and 8.

Conclusion

• Happy Thanksgiving!
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