Rational equations
A rational equation has the unknown in at least one denominator. The idea for solving it is always the same —clear the denominators by multiplying by their lowest common multiple— but the risk is also always the same: a value that looked like a solution may in fact make a denominator zero, and then it has to be rejected.
Domain and the basic method
Solve the following rational equation, clearly indicating its valid solutions: $\frac{3}{x} - \frac{1}{x+2} = \frac{5}{x(x+2)}$
Solve the following rational equation, clearly indicating its valid solutions:
$\frac{3}{x} - \frac{1}{x+2} = \frac{5}{x(x+2)}$
Determine the solution to the equation, paying attention to the values that make the denominators zero: $\frac{2x}{x+1} + \frac{1}{x-1} = 2$
Determine the solution to the equation, paying attention to the values that make the denominators zero:
$\frac{2x}{x+1} + \frac{1}{x-1} = 2$
Factorising the denominators before finding the l.c.m.
Find the solutions to the following equation: $\frac{4}{x-2} - \frac{3}{x+2} = \frac{1}{x^2-4}$
Find the solutions to the following equation:
$\frac{4}{x-2} - \frac{3}{x+2} = \frac{1}{x^2-4}$
Solve the rational equation by first factorising the denominators to find the lowest common multiple: $\frac{x}{x^2-4} + \frac{1}{x+2} = \frac{3}{x-2}$
Solve the rational equation by first factorising the denominators to find the lowest common multiple:
$\frac{x}{x^2-4} + \frac{1}{x+2} = \frac{3}{x-2}$
Beware of extraneous solutions
Find the solutions to the following equation, discarding any that turn out to be false (extraneous solutions): $\frac{2x+1}{x^2-x-6} - \frac{x}{x-3} = \frac{1}{x+2}$
Find the solutions to the following equation, discarding any that turn out to be false (extraneous solutions):
$\frac{2x+1}{x^2-x-6} - \frac{x}{x-3} = \frac{1}{x+2}$
Find the values of $x$ that satisfy the following equation: $\frac{x^2-x}{x^2-4x+3} = \frac{2x}{x-3} - 1$
Find the values of $x$ that satisfy the following equation:
$\frac{x^2-x}{x^2-4x+3} = \frac{2x}{x-3} - 1$
Problems with rational equations
Two 3D printers working together take $6$ hours to produce a full batch of parts for an engineering project. If they worked separately, printer A would take $5$ hours…
Two 3D printers working together take $6$ hours to produce a full batch of parts for an engineering project. If they worked separately, printer A would take $5$ hours less than printer B to complete the same batch. Set up a rational equation and determine how long printer A would take working alone.
A freight train must cover a distance of $300\text{ km}$. If it increased its average speed by $10\text{ km/h}$, it would take $1$ hour less to complete the whole…
A freight train must cover a distance of $300\text{ km}$. If it increased its average speed by $10\text{ km/h}$, it would take $1$ hour less to complete the whole journey. Set up a rational equation and calculate the average speed at which the train initially travels.
A graphic designer is creating a rectangular poster whose area must be exactly $600\text{ cm}^2$. If he increased the base of the poster by $10\text{ cm}$ and reduced…
A graphic designer is creating a rectangular poster whose area must be exactly $600\text{ cm}^2$. If he increased the base of the poster by $10\text{ cm}$ and reduced the height by $5\text{ cm}$, the total area would remain the same. Set up a rational equation and determine the original dimensions of the poster.
Parameters and more advanced equations
Given the rational equation with parameter $a$: $\frac{x+a}{x-1} + \frac{x-a}{x+1} = \frac{2x^2+2}{x^2-1}$ Determine for which values of parameter $a$ the equation has…
Given the rational equation with parameter $a$:
$\frac{x+a}{x-1} + \frac{x-a}{x+1} = \frac{2x^2+2}{x^2-1}$
Determine for which values of parameter $a$ the equation has infinitely many solutions, and for which values it has no solution.
Solve the following rational equation, first factorising the denominators to operate with the lowest common multiple:…
Solve the following rational equation, first factorising the denominators to operate with the lowest common multiple:
$\frac{1}{x^2-5x+6} + \frac{1}{x^2-7x+12} = \frac{2}{x^2-6x+8}$
Solve the following rational equation by applying the substitution $t = \frac{x^2+1}{x}$: $\frac{x^2+1}{x} + \frac{6x}{x^2+1} = 5$
Solve the following rational equation by applying the substitution $t = \frac{x^2+1}{x}$:
$\frac{x^2+1}{x} + \frac{6x}{x^2+1} = 5$