Linear Inequalities: Practice Problems and Methods

Linear inequalities and systems

Example: $3x - 4 < 5x + 2$ gives $-2x < 6$, so $x > -3$ (dividing by $-2$ reverses the sign).

For a system, solve each inequality and take the overlap on a number line. If there is no overlap, there is no solution. Clear fractions or decimals first by multiplying by a common denominator or by 10.

Equations and inequalities with absolute values

With two absolute values, as in $|x - 1| + |x - 3| < 4$, or with $x$ on the right, as in $|x - 2| = 2x + 1$, split into cases at the points where the expressions inside become 0. Always check that each solution lies in the range of its case.

Counting integer solutions; word problems

For a problem such as "$x > 2$ and $x < a$ have exactly 3 integer solutions", the integers must be $3, 4, 5$ but not $6$. Test the boundary values $a = 5$ and $a = 6$ directly to decide whether equality is allowed (the answer is $5 < a \leqq 6$).