Choose a variable that answers the question
An equation represents a relationship. In a word problem, define the unknown and its units before using it. If x represents a number of tickets, it is a nonnegative integer; if x represents a length, a negative solution must be rejected. The same algebraic operation can produce valid or invalid results depending on the context.
Also preserve restrictions that apply before simplification. Factoring and simplifying (x² − 9)/(x − 3) gives x + 3 for x ≠ 3; it does not make the original expression defined at x = 3. The restriction remains even when the expression is shorter. A comparison may focus precisely on this difference.
Exercise 1: check by substitution
Substitution is not a whole new method; it is a quick check in the original equation before relying on an answer.
If 4(x − 3) + 7 = 3x + 5, what is the value of x?
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The solution, step by step
10.
The method in action
- Expand: 4x − 12 + 7 = 3x + 5, or 4x − 5 = 3x + 5. Subtracting 3x and then adding 5 gives x = 10.
- In the original equation, the left side is 4 × 7 + 7 = 35; the right side is 30 + 5 = 35. Forgetting to multiply −3 by 4 would produce a different equation.
Exercise 2: reversing the inequality
Adding or subtracting the same value preserves the direction of an inequality. Multiplying or dividing by a negative number reverses it. If the factor’s sign is unknown, consider the cases before transforming the inequality; before dividing, also check that the divisor is nonzero.
Which values of x satisfy 7 − 3x < 16?
Choose an answer.
The solution, step by step
x > −3.
The method in action
- Subtracting 7 gives −3x < 9. Dividing by −3 reverses the inequality: x > −3.
- x = 0 satisfies the original inequality, 7 < 16. x = −4 gives 19 < 16, which is false. At the boundary −3, the two sides are equal, which is excluded by the strict inequality.
Exercise 3: keep both possibilities
A quadratic equation may have multiple solutions. Check all of them against the question instead of automatically keeping the positive root.
Select all values of x that satisfy x² − 5x + 6 = 0.
Select all correct answers.
The solution, step by step
2 and 3.
The method in action
- Factoring gives (x − 2)(x − 3) = 0. A zero product requires at least one factor to be zero: x = 2 or x = 3.
- 0 and 6 come from the coefficients without solving the equation. −2 gives 20 and −3 gives 30, so those values do not work. No additional domain restriction removes either root.
Exercise 4: read the input before calculating
The notation f(t) means apply the rule for f to input t. It is not multiplication f × t. If the input is an expression, substitute it in parentheses at every occurrence.
For all real x, f(x) = 2x² − 3. What is f(−2) − f(1)?
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The solution, step by step
6.
The method in action
- f(−2) = 2 × (−2)² − 3 = 5. f(1) = 2 − 3 = −1. The difference is 5 − (−1) = 6.
- The square applies to all of −2, so it is 4. Subtracting a negative value becomes addition. An answer of 4 overlooks this final sign.
The pitfall
The pitfall to avoid
Dividing by x without checking that x is nonzero or, in an inequality, knowing its sign is a mistake. A simplification can remove a solution, add a forbidden value, or reverse the comparison. Keep the conditions visible until you have your answer.
Move from technique to decision-making
After these exercises, choose one error and create a minimal variation. Replace a negative coefficient in an inequality with a positive one; make a boundary strict, then inclusive; require x to be an integer in an interval. Explain what changes in the answer and what stays the same in the calculations.
For a quantitative comparison, sometimes look for a difference or factorization instead of exact values. For a numeric entry, check the value in the original expression. For a multiple-answer question, check every permitted choice. Adapting to the format matters as much as the algebra.
Key takeaways
- Keep restrictions even after simplification.
- Multiplying or dividing by a negative number reverses an inequality.
- Check your answer in the original expression or equation.