Linear Equations and Inequalities
What you must understand
A linear equation in one variable can always be reduced to the form ax + b = c. Isolate the variable by undoing operations in reverse order: first clear parentheses and fractions, then collect variable terms on one side and constants on the other, then divide by the coefficient. For inequalities, the solution set is an interval, and the single most important rule is that multiplying or dividing both sides by a negative number reverses the inequality symbol. Compound inequalities of the form a < x < b describe an intersection; those joined by "or" describe a union. Absolute value equations split into two cases; absolute value inequalities of the form |x| < k become a double inequality, while |x| > k becomes a union of two rays.
Traps to watch
- Forgetting to flip the symbol when dividing by a negative coefficient, e.g. -2x > 6 gives x < -3, not x > -3.
- Distributing a negative sign to only the first term inside parentheses.
- Treating an absolute value as always positive on the right side: |x - 1| = -4 has no solution.
- Reporting a compound inequality as two separate answers when the question wants interval notation.
- Dividing by a variable expression without checking whether it could be zero.
Worked example
Solve 3(x - 4) <= 5x + 2. Distribute: 3x - 12 <= 5x + 2. Subtract 5x: -2x - 12 <= 2. Add 12: -2x <= 14. Divide by -2 and flip: x >= -7, or [-7, infinity).
Rule of thumb
Do the same thing to both sides, and the moment you multiply or divide by a negative, reverse the inequality.