A reliable method
Gather variable terms, isolate the relevant expression, then undo operations in reverse order. Check any result in the original statement.
The common trap
A square root denotes the nonnegative root. An absolute value equation may have two solutions; the question can request just one.
Make the connection
For an integer maximum, solve the boundary and round toward the feasible region. Test the last allowed integer and the next integer.
Worked example: choose the requested quantity
If 4x − 8 = 2x + 10, what is x? Subtract 2x from both sides and add the constant on the left. Divide by the remaining coefficient. The requested value is 9.
Contrast and diagnose an error
The equation |x − 3| = 4 has two solutions. What is the greater solution? The distance from x to the center equals the stated positive value. The greater solution lies that distance to the right of the center. The requested value is 7. A tempting answer is 1: Reverse center and distance in subtraction.
Transfer practice and check
If √(x + 4) = 3, what is x? Try it before reading on. Check: 5. Both sides are nonnegative, so square to get x plus the constant equal to the squared right side. Subtract the constant and verify the radicand.
If 4x − 8 = 2x + 10, what is x?
- Subtract 2x from both sides and add the constant on the left.
- Divide by the remaining coefficient.
- The requested value is 9.
Compare the other answer choices
Add rather than subtract the x coefficients.
Stop before dividing by the x coefficient.
Ignore the 2x term on the right.
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.
ACT Mathematics content