One Solution, None, or All - Practice 2

Decide what the final equation actually says instead of forcing one answer.

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Watch the lesson | Review: expand and solve

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Use valid expansion and reversible operations. A remaining nonzero coefficient gives one value. A true numerical statement after the variable cancels means all real numbers here; a false one means no solution. Never divide by zero.

Worked example

2(x + 3) = 2x + 9 reduces to 6 = 9, so there is no solution. In contrast, 2(x + 3) = 2x + 6 reduces to 6 = 6, so every real x works.

Three different final equations: 3x equals 12 gives the single solution 4; 6 equals 9 is false and gives no solution; 6 equals 6 is true and gives all real solutions after valid transformations. — These conclusions apply after valid simplification of the linear equations here, not from guessing a few test values.

Would you believe it?

A small change can switch an equation from no solutions to all real numbers: 2(x + 3) = 2x + 9 gives 6 = 9, while changing 9 to 6 gives 6 = 6. Read the final statement; never divide by zero.

OpenStax: types of linear equations

Question 1 Over the real numbers, classify 3(x + 2) = 3x + 6. Only x = 0 All real numbers No solution

Question 2 Over the real numbers, classify 4(x - 1) = 4x + 3. All real numbers Only x = 7 No solution

Question 3 Which conclusion follows from 2(x + 5) = 4x + 6? Only x = 2 All real numbers No solution

Question 4 Over the real numbers, classify -(x - 4) = 4 - x. No solution Only x = 4 All real numbers

Question 5 Which conclusion follows from 5(x - 2) = 5x - 10 + x? All real numbers No solution Only x = 0

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