Teach: Equations ③ Equations Containing Parentheses
Equations ③ Equations Containing Parentheses
When an equation contains parentheses, first remove the parentheses and then solve it. The distributive property is used more often, so carefully handle changes in signs.
Steps for Removing Parentheses
Use the following steps to solve an equation containing parentheses.
① Remove the parentheses using the distributive property
② Combine like terms (if necessary)
③ Transpose variable terms to the left and constant terms to the right
④ Simplify both sides and divide by the coefficient
Example ①
Solve the following equation: 3(x − 2) = 9
Answer:
① Remove the parentheses: 3×x − 3×2 = 9 → 3x − 6 = 9
② Transpose: 3x = 9 + 6 → 3x = 15
③ Divide both sides by 3: x = 5
Solve the following equation: 3(x − 2) = 9
Answer:
① Remove the parentheses: 3×x − 3×2 = 9 → 3x − 6 = 9
② Transpose: 3x = 9 + 6 → 3x = 15
③ Divide both sides by 3: x = 5
Example ②
Solve the following equation: 5x − 2(x − 4) = 17
Answer:
① Remove the parentheses: −2(x − 4) = −2x + 8
Therefore, 5x − 2x + 8 = 17
② Combine like terms: 3x + 8 = 17
③ Transpose: 3x = 17 − 8 → 3x = 9
④ Divide both sides by 3: x = 3
Solve the following equation: 5x − 2(x − 4) = 17
Answer:
① Remove the parentheses: −2(x − 4) = −2x + 8
Therefore, 5x − 2x + 8 = 17
② Combine like terms: 3x + 8 = 17
③ Transpose: 3x = 17 − 8 → 3x = 9
④ Divide both sides by 3: x = 3
Patterns with Parentheses on Both Sides
When both the left and right sides have parentheses, first remove the parentheses on each side, then transpose the terms.
Example ③
Solve the following equation: 2(x + 3) = 4(x − 1)
Answer:
① Remove the parentheses: 2x + 6 = 4x − 4
② Transpose: 2x − 4x = −4 − 6
③ Simplify: −2x = −10
④ Divide both sides by −2: x = 5
Solve the following equation: 2(x + 3) = 4(x − 1)
Answer:
① Remove the parentheses: 2x + 6 = 4x − 4
② Transpose: 2x − 4x = −4 − 6
③ Simplify: −2x = −10
④ Divide both sides by −2: x = 5
Key Points
- For equations containing parentheses: ① distribute to remove them ② combine like terms ③ transpose ④ divide by the coefficient
- When the expression before the parentheses is “−”, all signs inside change when the parentheses are removed
- −2(x − 4) becomes “−2x + 8.” Do not forget −2×(−4) = +8
- The same procedure is used when both sides have parentheses
Practice Problems
- Solve the following: 2(x + 5) = 16
- Solve the following: 4x − 3(x − 2) = 11
- Solve the following: 3(2x − 1) = 5(x + 1)
Answers and Explanations
- Answer: x = 3
Explanation: 2x + 10 = 16 → 2x = 6 → x = 3. - Answer: x = 5
Explanation: −3(x−2) = −3x + 6. The expression becomes 4x − 3x + 6 = 11 → x + 6 = 11 → x = 5. - Answer: x = 8
Explanation: 6x − 3 = 5x + 5 → 6x − 5x = 5 + 3 → x = 8.
Previous and next lessons in this course
- Previous lesson: Teach: Equations ② Solving Equations Using Transposition
- Next lesson: Fix the mistake: Solving Linear Equations in One Variable