A reliable method
Distribute first, keep the entire equation balanced, and apply restrictions after solving. For an absolute-value equation, split into positive and negative cases.
The common trap
A real-number solution need not satisfy a requirement that the variable be an integer. Squaring can create extraneous candidates and division can discard zero solutions.
Make the connection
Use substitution in the original statement as a final check. For systems, ensure both equations hold; satisfying only one condition is not enough.
Contrast the cases
The equation |x−2|=5 has candidates 7 and −3. If the additional condition says x>0, only 7 survives. Without that condition, rejecting −3 is an unjustified extra assumption. By contrast, 3x<12 gives x<4, so positive integer inputs are 1,2,3. Equality would include 4 only if the original symbol allowed it. Keep the real solution set and the context restriction as separate steps.
Transfer and explain
Transfer: solve |x+1|=4 and then impose x<0. The algebra gives 3 and −5; the condition selects −5. Check it in the original absolute-value expression.
If 2x + 9 = 23, what is 3x − 2?
- Subtract 9 and divide by 2 to obtain x = 7.
- Substitute into the requested expression, not just x.
- 3(7) − 2 = 19.
Compare the other answer choices
Omits the constant in the requested expression.
Reverses the sign of the requested constant.
Adds rather than subtracts the constant when isolating x.
Ignores the original constant b.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference