Algebra / Linear Inequalities
Linear Inequalities
Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Solving them is similar to solving linear equations, except the inequality symbol reverses when both sides are multiplied or divided by a negative number.
Least you need to know
- The solution to a one-variable inequality is usually a range of values.
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- A two-variable linear inequality describes a half-plane whose boundary is a line.
Key notation
- $x<5$ — All values of $x$ less than 5
- $y\ge 2x-1$ — All ordered pairs on or above the boundary line $y=2x-1$
Worked example
- Consider $-3x+4<16$.
- Subtract 4 from both sides: $-3x<12$.
- Divide by $-3$ and reverse the inequality: $x>-4$.
Common mistakes
- Reverse the inequality symbol when multiplying or dividing by a negative number.
- An inequality with $\le$ or $\ge$ includes the boundary value.
- A point satisfies a two-variable inequality only if substituting its coordinates makes the inequality true.
How to recognize it
- The problem uses $<$, $>$, $\le$, or $\ge$.
- The answer may be a range, a greatest or least possible integer, or an ordered pair that satisfies an inequality.
- Context questions may use phrases such as at least, at most, fewer than, or no more than.
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