Lesson 1 · from §35 of the 1910 text
Equations with Parentheses
Removing parentheses is easy and follows simple rules, but it's also very easy to pick up careless habits that are hard to break. The way to be accurate is to take special care with every parenthesis that has a minus sign in front of it.
Example. Solve 5(2x - 1) - 3(4x - 6) = 7. \begin{array}{rll} (10x - 5) - (12x - 18) &= 7 & \text{multiply by 5 and by 3} \\ 10x - 5 - 12x + 18 &= 7 & \text{remove parentheses (watch the minus!)} \\ -2x + 13 &= 7 & \text{combine} \\ -2x &= -6 & \text{transpose} \\ x &= 3 & \text{divide by } -2 \end{array} Check: 5(6 - 1) - 3(12 - 6) = 25 - 18 = 7. ✓
Quick check
Solve 3(x + 4) - 2(x - 1) = 20.
When the square cancels
Sometimes the square of the unknown appears and then vanishes. Solve 4 + (n - 3)(n - 5) = 15 - (7 - n)(2 + n): \begin{array}{rl} 4 + (n^2 - 8n + 15) &= 15 - (14 + 5n - n^2) \\ 4 + n^2 - 8n + 15 &= 15 - 14 - 5n + n^2 \\ 19 - 8n &= 1 - 5n \quad \text{(subtract } n^2 \text{ from each side)} \\ -3n &= -18 \\ n &= 6 \end{array} Check: 4 + (3)(1) = 15 - (1)(8), or 7 = 7. ✓
Quick check
Solve (x + 1)(x + 2) = x^2 + 11.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Remove parentheses carefully, then solve and check.
Solve 5(x - 1) = 30. What is x?
Solve 3 + 2(x - 3) = 1. What is x?
Solve 7(3x - 2) + 11 = 60. What is x?
Solve 4(2x - 5) + 15 = 3(x + 10). What is x?
Solve 12y - 2(4y - 7) - 16 = 0. What is y?
Solve 9y - 3(2y - 4) = 2(5 - 4y) + 2. What is y?
Solve 4 - 2(4y - 3) = 3(y - 5). What is y?
Solve 7(y - 3) - 2(4 + y) = 9. What is y?
Solve 5(n - 7) + 24 + 4n = 0. What is n?
Solve 5n - 9(2n + 4) = 2(n - 9). What is n?
Solve 7n - 12 - 2(n - 5) = n - 19. What is n?
Solve 4(2n - 7) - 3(4n - 8) + 4 = 2n - 3. What is n?
Solve 3h - 2(4h + 8) = 3h - 24. What is h?
Solve 5(3h + 1) - 7h = 3(h - 7) + 4. What is h?
Solve (h - 2)(h - 5) = (h + 3)(h + 2). What is h?
Solve (h + 4)(h + 3) - (h + 2)(h + 1) - 42 = 0. What is h?
Solve (x + 4)(x + 6) = (x + 18)(x + 13). What is x?
Solve (k - 7)(5 + k) - (k - 5)(k + 7) + 5 = 0. What is k?
Solve (2x - 5)(4x - 7) = 8x^2 + 52. What is x?
Solve (3y + 5)(4y + 7) - (2y + 3)(6y + 11) - 2 = 0. What is y?
Solve (n + 3)(6n + 5) - (2n + 4)(3n - 8) = 38. What is n?
Solve (x + 3)^2 - (x + 5)^2 = -40. What is x?
Solve (x + 2)^2 - (x - 4)^2 + 48 = 0. What is x?