Inequality Solver

An inequality is a mathematical statement that compares two expressions using one of the symbols:

• $<$ (less than)
• $>$ (greater than)
• $\leq$ (less than or equal to)
• $\geq$ (greater than or equal to)

Unlike equations (which ask "what values make two sides equal?"), inequalities ask "what values make one side larger (or smaller) than the other?"

For example, the inequality:

$2x - 5 > 3$

asks: for which values of $x$ is $2x - 5$ greater than $3$?

The solution to an inequality is typically a range of values (an interval), not a single number. Solutions are often expressed in interval notation:

• $(a, b)$: all values strictly between $a$ and $b$
• $[a, b]$: all values from $a$ to $b$, inclusive
• $(-\infty, a) \cup (b, \infty)$: all values less than $a$ or greater than $b$

Inequalities are fundamental in optimization, constraint problems, and determining domains and ranges of functions.

1. Linear Inequalities

Solve like a linear equation, with one critical rule: flipping the inequality sign when multiplying or dividing by a negative number.

Example: Solve $2x - 5 > 3$

1. Add 5: $2x > 8$
2. Divide by 2: $x > 4$

Solution: $(4, \infty)$

Example with sign flip: Solve $-3x + 6 \leq 12$

1. Subtract 6: $-3x \leq 6$
2. Divide by $-3$ (flip!): $x \geq -2$

2. Quadratic Inequalities

Solve the corresponding equation first, then test intervals.

Example: Solve $x^2 - 4x - 5 > 0$

1. Factor: $(x - 5)(x + 1) > 0$
2. Critical points: $x = -1$ and $x = 5$
3. Test intervals:
   • $x < -1$: $(-)(-) = (+) > 0$ ✓
   • $-1 < x < 5$: $(-)(+) = (-) < 0$ ✗
   • $x > 5$: $(+)(+) = (+) > 0$ ✓

Solution: $(-\infty, -1) \cup (5, \infty)$

3. Rational Inequalities

Find where the numerator and denominator are zero (critical points), then test sign in each interval. Never multiply both sides by an expression that could be negative.

4. Absolute Value Inequalities

• $|x| < a$ means $-a < x < a$
• $|x| > a$ means $x < -a$ or $x > a$

5. Sign Chart Method

For polynomial/rational inequalities, build a sign chart showing the sign of each factor in each interval.

• Forgetting to flip the inequality sign: When you multiply or divide both sides by a negative number, you must reverse the direction of the inequality.
• Including critical points incorrectly: For strict inequalities ($<$, $>$), critical points are NOT included. For $\leq$ or $\geq$, they are.
• Multiplying by a variable without considering its sign: If you multiply both sides by $x$, you must consider cases where $x > 0$ and $x < 0$ separately.
• Treating compound inequalities incorrectly: For $a < f(x) < b$, solve both parts simultaneously, not independently.
• Writing solution in wrong notation: Use parentheses for strict inequalities and brackets for inclusive ones.