MENUEquation and Inequality Solver with Steps
Type an equation or an inequality in one variable. The solver finds the solution and shows every step, the way it is done at school: linear, quadratic, polynomial and rational equations, and linear, quadratic, polynomial, rational and double inequalities.
Examples: $3(x - 2) + 4 = 2x - 1$ $x^2 - 5x + 6 = 0$ $2x^2 + 3x - 2 = 0$ $\frac{x+1}{x-2} = 3$ $x^4 - 5x^2 + 4 = 0$ $5 - 3x \le 11$ $x^2 - 5x + 6 < 0$ $\frac{x-1}{x+2} \ge 0$ $1 < 2x + 3 \le 7$
Solution
Step-by-step solution
How to enter a problem
=makes an equation;<,>,<=and>=(or ≤ and ≥) make an inequality. A double inequality such as1 < 2x + 3 <= 7is solved too.*is multiplication; it can be left out:2x,3(x - 2),(x + 1)(x - 2).x^2is $x^2$.(x+1)/(x-2)is $\frac{x+1}{x-2}$: put the numerator and the denominator in brackets.- Decimals are written with a point:
0.5x + 1.2 = 3. - The variable is $x$, or the one letter of the problem:
2t + 1 = 5is solved for $t$.
Linear equations
Clear the fractions by multiplying both sides by the common denominator, and expand the brackets. Then move the terms with $x$ to one side and the numbers to the other; a term changes its sign when it moves across the equals sign. Collect like terms and divide by the coefficient of $x$:
$3(x - 2) + 4 = 2x - 1 \quad\Rightarrow\quad 3x - 2 = 2x - 1 \quad\Rightarrow\quad 3x - 2x = -1 + 2 \quad\Rightarrow\quad x = 1$
More: Solving linear equations.
Quadratic equations
A quadratic equation $ax^2 + bx + c = 0$ is solved with the discriminant $D = b^2 - 4ac$:
$x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}$
If $D > 0$, there are two roots; if $D = 0$, one root; if $D < 0$, no real roots. Two cases are quicker: when $c = 0$, take $x$ out of the brackets, $x(ax + b) = 0$; when $b = 0$, find $x^2 = -\frac{c}{a}$ and take the square root. When $a = 1$ and the roots are whole numbers, look for two numbers whose product is $c$ and whose sum is $b$: $x^2 - 5x + 6 = (x - 2)(x - 3)$.
More: Quadratic equations.
Polynomial equations
A polynomial equation of a higher degree is solved by factoring: a product is zero when one of its factors is zero. Look for whole roots among the divisors of the constant term. A biquadratic equation $ax^4 + bx^2 + c = 0$ becomes a quadratic one with $t = x^2$.
More: Biquadratic equations, Cubic and quartic equations.
Rational equations
First find the values that make a denominator zero: they cannot be solutions. Then multiply both sides by the common denominator and solve the equation without fractions. At the end, drop every root that makes a denominator zero.
Inequalities
A linear inequality is solved like an equation, with one rule: multiplying or dividing both sides by a negative number reverses the inequality sign. $-3x \le 6 \;\Rightarrow\; x \ge -2$.
Other inequalities are solved with a sign chart. Move all terms to one side and write them as one fraction; do not multiply by an expression with $x$, because its sign is not known. Factor the numerator and the denominator. The expression can change its sign only where a factor is zero, so find the sign between those points and choose the intervals where the inequality holds. The zeros of the denominator never belong to the answer.
More: Linear inequalities, Quadratic inequalities, Polynomial and rational inequalities.
