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lesson 3.4 solving complex 1-variable equations answer key

lesson 3.4 solving complex 1-variable equations answer key

3 min read 19-12-2024
lesson 3.4 solving complex 1-variable equations answer key

I cannot provide a complete answer key for "Lesson 3.4 Solving Complex 1-Variable Equations" because I don't have access to the specific textbook or curriculum you're using. The complexity of the equations and the methods taught will vary depending on the source.

However, I can give you a comprehensive guide on how to solve complex one-variable equations, covering techniques you'll likely encounter in such a lesson. This will enable you to tackle the problems in your assignment yourself.

This guide will walk you through various methods for solving complex equations with one variable. Remember, the key is to isolate the variable on one side of the equation.

Types of Complex Equations and Solution Strategies

Here are some common types of complex one-variable equations and strategies to solve them:

1. Equations with Variables on Both Sides

Example: 3x + 5 = 2x - 7

  • Strategy: Combine like terms by subtracting the smaller variable term from both sides. Then, isolate the variable by using inverse operations (addition/subtraction, then multiplication/division).

Solution:

  1. Subtract 2x from both sides: x + 5 = -7
  2. Subtract 5 from both sides: x = -12

2. Equations with Parentheses

Example: 2(x + 3) - 5 = 11

  • Strategy: Distribute the number outside the parentheses to each term inside. Then, simplify and solve as usual.

Solution:

  1. Distribute the 2: 2x + 6 - 5 = 11
  2. Combine like terms: 2x + 1 = 11
  3. Subtract 1 from both sides: 2x = 10
  4. Divide both sides by 2: x = 5

3. Equations with Fractions

Example: (x/3) + 4 = 7

  • Strategy: Eliminate fractions by multiplying both sides by the least common denominator (LCD) of all the fractions. This will clear the fractions.

Solution:

  1. The LCD is 3. Multiply both sides by 3: 3 * (x/3) + 3 * 4 = 3 * 7
  2. Simplify: x + 12 = 21
  3. Subtract 12 from both sides: x = 9

Example (more complex fractions): (2x + 1)/5 - (x - 3)/2 = 1

  • Strategy: Find the LCD (10 in this case). Multiply both sides by the LCD, then distribute and solve.

Solution:

  1. Multiply both sides by 10: 10 * [(2x + 1)/5 - (x - 3)/2] = 10 * 1
  2. Distribute: 2(2x + 1) - 5(x - 3) = 10
  3. Simplify and solve: 4x + 2 - 5x + 15 = 10 => -x + 17 = 10 => x = 7

4. Equations with Decimals

Example: 0.5x + 2.3 = 5.8

  • Strategy: You can solve directly with decimals, or multiply the entire equation by a power of 10 (e.g., 10) to eliminate the decimals and work with integers, making calculations easier.

Solution (eliminating decimals):

  1. Multiply by 10: 5x + 23 = 58
  2. Solve: 5x = 35 => x = 7

5. Equations with Absolute Value

Example: |2x - 1| = 5

  • Strategy: Set up two separate equations: 2x - 1 = 5 and 2x - 1 = -5. Solve each equation individually.

Solution:

  1. 2x - 1 = 5: 2x = 6 => x = 3
  2. 2x - 1 = -5: 2x = -4 => x = -2 Therefore, the solutions are x = 3 and x = -2.

6. Equations with Square Roots

Example: √(x + 2) = 4

  • Strategy: Isolate the square root, then square both sides to eliminate the radical. Check your solution to ensure it doesn't introduce extraneous solutions (solutions that don't work in the original equation).

Solution:

  1. Square both sides: (√(x + 2))² = 4²
  2. Simplify: x + 2 = 16
  3. Solve: x = 14
  4. Check: √(14 + 2) = √16 = 4. The solution is valid.

Troubleshooting and Common Mistakes

  • Order of Operations (PEMDAS/BODMAS): Always follow the order of operations when simplifying expressions.
  • Sign Errors: Be meticulous with positive and negative signs, especially when distributing or combining like terms.
  • Fractions: Remember to multiply all terms by the LCD when clearing fractions.
  • Checking Your Answer: Always plug your solution back into the original equation to verify it's correct.

Remember to consult your textbook or class notes for specific examples and methods taught in Lesson 3.4. Use this guide to help you understand the techniques and work through the problems in your assignment. If you're still stuck on a particular problem, provide the problem itself, and I can guide you through the solution.

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