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1. Learning Objectives
- Synthesize all methods learned to solve complex multi-step problems.
- Review the connection between linear and non-linear models.
- Apply mathematical reasoning to evaluate solutions for real-world accuracy.
2. Unit Summary Table
Linear (3.1): One solution ($x = k$).
Systems (3.2): Intersection of two lines ($x, y$).
Non-linear (3.3): Multiple cases ($|x|$) or roots ($x^2$).
Applications (3.4): Modeling real scenarios into math.
Click the button below each question to reveal the explanation.
1. Solve for x: 5(x - 2) = 3(x + 4)
5x - 10 = 3x + 12 → 2x = 22 → x = 11. Ans: 11
2. Solve the system: x + y = 12 and y = x
Substitute x for y: x + x = 12 → 2x = 12 → x = 6. Since y = x, y = 6. Ans: (6, 6)
3. Solve: |2x + 1| = 5
Case 1: 2x+1=5 → x=2. Case 2: 2x+1=-5 → x=-3. Ans: 2, -3
4. Solve: x² - 25 = 0
x² = 25 → x = ±5. Ans: 5, -5
5. A number is doubled and then 7 is added. The result is 15. Find it.
2x + 7 = 15 → 2x = 8 → x = 4. Ans: 4
6. Find the discriminant of 2x² - 4x + 2 = 0
D = (-4)² - 4(2)(2) = 16 - 16 = 0. Ans: 0 (One Real Root)
7. Solve: 4^x = 64
4^x = 4^3, so x = 3. Ans: 3
8. Solve the system: x + y = 2 and x - y = 0
Add: 2x = 2 → x=1. Sub: 1-y=0 → y=1. Ans: (1, 1)
9. Solve: √x = 9
Square both sides: x = 9² = 81. Ans: 81
10. If |x - 2| = 0, what is x?
x - 2 = 0 → x = 2. Ans: 2
11. A rectangle has perimeter 40. Length is 3 times width. Find width.
2(w + 3w) = 40 → 8w = 40 → w = 5. Ans: 5
12. Solve: x² + 6x + 9 = 0
(x+3)² = 0 → x = -3. Ans: -3
13. Solve: |x| + 5 = 5
|x| = 0 → x = 0. Ans: 0
14. Solve: 3^x = 1/3
3^x = 3^(-1) → x = -1. Ans: -1
15. How many solutions for parallel lines?
Parallel lines never cross. Ans: No Solution
16. If x/4 = 5, find x.
Multiply by 4: x = 20. Ans: 20
17. Solve the system: x = 10 and y = x - 5
Sub x=10: y = 10 - 5 = 5. Ans: (10, 5)
18. Factor: x² - x - 6 = 0
(x-3)(x+2) = 0 → x=3, -2. Ans: 3, -2
19. If y = 2x and 2x + y = 8, solve for x.
Sub y with 2x: 2x + 2x = 8 → 4x = 8 → x = 2. Ans: 2
20. What is the value of x in 7^x = 1?
Any non-zero base to power 0 equals 1. Ans: 0