Solving Linear Inequalities

Solve inequalities exactly like equations, with one exception: multiplying or dividing both sides by a negative flips the direction of the sign.

−3x + 7 > 22
−3x > 15
x < −5 ← flipped, because we divided by −3

Adding or subtracting a negative does *not* flip anything. Only multiplying or dividing by one does.

Write the answer as a graph (open circle for < or >, closed for ≤ or ≥) or in interval notation: x < −5 is (−∞, −5).

Test one value from your answer in the original. If it works, you probably got the direction right.

Order is preserved by addition and by multiplication by a positive, and reversed by multiplication by a negative. The reversal is a reflection of the number line through 0, and reflections reverse order.

Solution sets are intervals, so interval notation is the natural output: a parenthesis for a strict bound or an infinite end, a bracket for an included endpoint. x ≤ 4 is (−∞, 4], and ∞ never takes a bracket because it is not a number being included.

The rule that a variable multiplier requires known sign has real consequences. You may not multiply 3/x < 5 through by x, because x may be negative and the direction would depend on the case. Rational inequalities are instead solved by moving everything to one side and testing the sign on each interval between the zeros and undefined points — a technique that carries directly into quadratic and polynomial inequalities.