Solving Linear Inequalities

An inequality compares quantities that may not be equal. The symbols mean less than and greater than. The symbols = include equality. A solution is any value that makes the inequality true, so an inequality often has infinitely many solutions.

First decide whether a number is a solution by substituting it. For x + 3 > 8, x = 6 works because 9 > 8, but x = 4 does not because 7 > 8 is false.

Solve inequalities using the balance idea from equations. Add or subtract the same number on both sides without changing the inequality direction. Example: x - 5 <= 9. Add 5 to both sides to get x 12, divide by -3 and reverse > to <: x = 13 gives 4x >= 20, so x >= 5. With rational coefficients, the same rules apply to fractions and decimals.

Compound inequalities combine conditions. The word and means both conditions must be true, as in 2 < x <= 7. The word or means either condition may be true, as in x = 5.

To model a situation, define a variable and translate phrases carefully: at least means >=, no more than means , and fewer than means <. Solve, graph, and interpret the answer with units. Always test a value from the solution region in the original inequality.