Inequalities Solver
Solve Simple Linear Inequalities
A linear inequality involves a linear expression, typically of the form $ax + b < c$, $ax + b > c$, $ax + b \le c$, or $ax + b \ge c$.
Enter Linear Inequality: $ax + b \text{ (op) } c$
Solve Inequalities Involving a Modulus
A modulus inequality involves the absolute value of an expression, e.g., $|ax + b| < c$.
Key properties:
- If $|X| < k$ (and $k > 0$), then $-k < X < k$.
- If $|X| > k$ (and $k > 0$), then $X < -k$ or $X > k$.
- If $|X| \le k$ (and $k > 0$), then $-k \le X \le k$.
- If $|X| \ge k$ (and $k > 0$), then $X \le -k$ or $X \ge k$.
Enter Modulus Inequality: $|ax + b| \text{ (op) } c$
Solve Quadratic Inequalities
A quadratic inequality involves a quadratic expression, e.g., $ax^2 + bx + c < 0$. To solve it, we typically find the roots of the quadratic equation $ax^2 + bx + c = 0$ and then test intervals or use the shape of the parabola.
Enter Quadratic Inequality: $ax^2 + bx + c \text{ (op) } 0$
Solve Inequalities Involving Square Functions
These are inequalities like $(px+q)^2 < k$ or $(px+q)^2 > k$.
Enter Square Function Inequality: $(px+q)^2 \text{ (op) } k$
Solve Rational Inequalities
A rational inequality involves a quotient of two polynomials, e.g., $\frac{P(x)}{Q(x)} > 0$.
Enter Rational Inequality: $\frac{ax+b}{cx+d} \text{ (op) } 0$
This solver handles linear numerator and linear denominator compared to 0.
What is an Inequality?
An inequality is a mathematical statement that compares two expressions using an inequality symbol. These symbols indicate that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.
Common inequality symbols:
- $<$ : Less than (e.g., $x < 5$ means $x$ is any number less than 5)
- $>$ : Greater than (e.g., $y > -2$ means $y$ is any number greater than -2)
- $\le$ : Less than or equal to (e.g., $a \le 10$ means $a$ can be 10 or any number less than 10)
- $\ge$ : Greater than or equal to (e.g., $b \ge 0$ means $b$ can be 0 or any number greater than 0)
- $\ne$ : Not equal to (e.g., $z \ne 3$ means $z$ can be any number except 3)
Solving an inequality means finding all the values of the variable that make the inequality true. The solution is often expressed as a set of numbers or an interval.
Simple Rules for Inequalities
When working with inequalities, certain operations can be performed on both sides without changing the truth of the inequality, similar to equations. However, there's one crucial difference when multiplying or dividing by negative numbers.
- Adding or Subtracting: You can add or subtract the same number (or expression) from both sides of an inequality, and the inequality symbol remains unchanged.
If $a < b$, then $a+c < b+c$ and $a-c < b-c$.
Example: Given $x - 3 > 7$. Add 3 to both sides: $x - 3 + 3 > 7 + 3 \implies x > 10$. - Multiplying or Dividing by a Positive Number: You can multiply or divide both sides of an inequality by the same positive number, and the inequality symbol remains unchanged.
If $a < b$ and $c > 0$, then $ac < bc$ and $a/c < b/c$.
Example: Given $2x \le 8$. Divide both sides by 2: $2x/2 \le 8/2 \implies x \le 4$. - Multiplying or Dividing by a Negative Number: If you multiply or divide both sides of an inequality by the same negative number, you must reverse the direction of the inequality symbol.
If $a < b$ and $c < 0$, then $ac > bc$ and $a/c > b/c$.
Example: Given $-3x < 12$. Divide both sides by -3 and reverse the symbol: $-3x/(-3) > 12/(-3) \implies x > -4$.