Linear Equations and Inequalities
Undo operations in reverse order
A linear equation states that two expressions are equal, and solving means isolating the variable. Work in reverse order of operations: clear parentheses with the distributive property, combine like terms on each side, collect variable terms on one side and constants on the other, then divide by the coefficient. When fractions appear, multiply every term by the least common denominator to clear them in one step.
Inequalities use the same steps with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality symbol. Compound inequalities such as −3 < 2x + 1 ≤ 7 are handled as two constraints applied to the same expression, so the solution is a single interval.
Traps to watch: distributing a negative to only the first term inside parentheses; dropping a term when moving it across the equals sign; and forgetting to flip the symbol. Also distinguish equations with no solution (you reach 0 = 5) from identities with all real numbers as solutions (you reach 0 = 0).
Worked example: 3(x − 4) = 2x + 5. Distribute: 3x − 12 = 2x + 5. Subtract 2x from both sides: x − 12 = 5. Add 12: x = 17. Check by substitution: 3(17 − 4) = 39 and 2(17) + 5 = 39, so the solution is confirmed.
Rule of thumb: whatever operation you perform, perform it on both sides, and always verify by substituting your answer back into the original statement.