10 Inequalities
An inequality is a statement of the form \(2x+3>4\), or \(x\leq 7\), or \(2x-1\geq 3x+2\), built with one of the symbols \(<\), \(>\), \(\leq \), \(\geq \).
Solving one is like solving an equation with a single crucial difference, and it is worth stating before anything else.
Theorem 10.1 (The rule that governs everything here). Adding or subtracting the same quantity from both sides leaves an inequality unchanged. So does multiplying or dividing both sides by a positive number. But multiplying or dividing both sides by a negative number reverses the inequality: \[a<b \quad \text {and}\quad c<0 \quad \Longrightarrow \quad ac>bc .\]
Note 10.2. Check it once on numbers and it will stay checked. From \(2<5\), multiplying by \(-1\) gives \(-2\) and \(-5\), and \(-2>-5\). On the number line, multiplying by a negative reflects every point through the origin, and a reflection reverses left and right.
Almost every wrong answer in this chapter comes from forgetting this rule — and from a second consequence of it that is easier to miss. You may not multiply both sides by an expression whose sign you do not know. That is why cross-multiplying is forbidden for rational inequalities: in \(\frac {x}{x+2}\geq 3\) the quantity \(x+2\) is positive for some \(x\) and negative for others, so there is no single direction the inequality would take. The safe route is always to move everything to one side and examine the sign of the result, which is what the tables below do.
Four types occur in this course:
- 1.
- Linear inequalities, \(ax+b\geq 0\);
- 2.
- absolute value inequalities, \(\left |ax+b\right |<k\);
- 3.
- quadratic inequalities, \(ax^{2}+bx+c\leq 0\);
- 4.
- rational inequalities, \(\frac {ax+b}{cx+d}\leq k\).
10.2 Absolute Value Inequalities
10.3 Quadratic Inequalities
10.4 Rational Inequalities
10.5 Practice Problems
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