One Solution, None, or All - More Practice

Read the equation left after valid simplification. Do not guess from a few test values.

All worksheets

Watch the lesson | One solution, none, or all: first practice

5 questions0m 0s

Learn the skill

Expand each bracket and combine like terms. If a nonzero coefficient of x remains, solve for one value. If x cancels, a true numerical statement means all real numbers here; a false statement means no solution. Never divide by zero.

Worked example

3(x - 2) = 3x + 1 reduces to -6 = 1, so there is no solution. But 3(x - 2) = 3x - 6 reduces to -6 = -6, so every real number works. Changing one constant changes the conclusion.

Three final equations show three possibilities: three x equals twelve gives only four; six equals nine is false and gives no solution; six equals six is true and gives all real numbers after reversible simplification. — A true or false numerical statement follows from algebra applied to the original equation.

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 -2(x + 3) = -2x + 6. Only x = 0 All real numbers No solution

Question 2 Over the real numbers, classify 5 - (x + 2) = 3 - x. All real numbers No solution Only x = 3

Question 3 Which conclusion follows from 3(x - 1) = x - 9? No solution Only x = -3 All real numbers

Question 4 Which conclusion follows from 2(x + 4) = 3x + 8? Only x = 0 All real numbers No solution

Question 5 Over the real numbers, classify -3(2x - 1) = 3 - 6x. No solution Only x = 1 All real numbers

Review your work

0correct

Complete the questions to review each answer.

View dashboard