Rational equations
A group of students from a high school in La Vega rents a bus to go to Samaná and splits the cost among everyone. If more classmates join, each person's share goes down. The number of students ends up in the denominator of the share, so to find out how many there were at the start you have to solve a rational equation. At the end of the page you will solve that problem.
Dominio y método básico
The steps are always the same: domain, LCM, multiply, solve, and check.
If the equation is an equality between two fractions, you can also cross-multiply. This is equivalent to multiplying by the product of the two denominators: $\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow ad=bc$. The domain rule still applies.
Look at where the two curves $y=\dfrac3x-\dfrac1{x+2}$ and $y=\dfrac{5}{x(x+2)}$ intersect. The x-coordinate of the intersection is the solution of $\dfrac{3}{x}-\dfrac{1}{x+2}=\dfrac{5}{x(x+2)}$.
Solve the following rational equation, clearly indicating its valid solutions: $\frac{3}{x} - \frac{1}{x+2} = \frac{5}{x(x+2)}$
Solve the following rational equation, clearly indicating its valid solutions:
$\frac{3}{x} - \frac{1}{x+2} = \frac{5}{x(x+2)}$
Find the values of $x$ that satisfy the following equality: $\frac{x+3}{x-1} = \frac{x-2}{x+1}$
Find the values of $x$ that satisfy the following equality:
$\frac{x+3}{x-1} = \frac{x-2}{x+1}$
Solve the rational equation by first putting it over a common denominator: $\frac{1}{x} + \frac{1}{2x} + \frac{1}{3x} = \frac{11}{6}$
Solve the rational equation by first putting it over a common denominator:
$\frac{1}{x} + \frac{1}{2x} + \frac{1}{3x} = \frac{11}{6}$
Factoring the denominators before the LCM
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:
$\frac{x}{x^2-4} + \frac{1}{x+2} = \frac{3}{x-2}$
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:
$\frac{1}{x^2-5x+6} + \frac{1}{x^2-7x+12} = \frac{2}{x^2-6x+8}$
Simplifying with special products
Sometimes a fraction can be simplified before operating. To do that, recognize the special products and factoring:
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$a^2-b^2=(a-b)(a+b)$
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$a^3-b^3=(a-b)(a^2+ab+b^2)$
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$a^3+b^3=(a+b)(a^2-ab+b^2)$
When simplifying, the original domain is kept: even if the factor $(x-3)$ disappears, $x=3$ is still forbidden.
Calculate the real values of $x$ for which the following expression holds: $\frac{x^2-9}{x-3} = 6$
Calculate the real values of $x$ for which the following expression holds:
$\frac{x^2-9}{x-3} = 6$
Solve the equation by simplifying the algebraic fractions: $\frac{x+1}{x-1} - \frac{x-1}{x+1} = \frac{4}{x^2-1}$
Solve the equation by simplifying the algebraic fractions:
$\frac{x+1}{x-1} - \frac{x-1}{x+1} = \frac{4}{x^2-1}$
Determine all the real solutions to the equation, simplifying the algebraic fractions using algebraic identities before operating:…
Determine all the real solutions to the equation, simplifying the algebraic fractions using algebraic identities before operating:
$\frac{x^3-8}{x-2} = \frac{x^3+1}{x+1} + 3$
Watch out for extraneous solutions
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:
$\frac{2x}{x+1} + \frac{1}{x-1} = 2$
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: work and speed
Steps for setting up a problem: define the unknown with units, write the other quantities in terms of it, set up the equation, solve it, discard whatever makes no sense (negative times or speeds), and answer with units.
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 motorboat travels along a river, covering $24\text{ km}$ downstream and then returning to the starting point upstream. The total navigation time was $5\text{ hours}$.…
A motorboat travels along a river, covering $24\text{ km}$ downstream and then returning to the starting point upstream. The total navigation time was $5\text{ hours}$. If the speed of the river current is $2\text{ km/h}$, set up a rational equation to calculate the speed of the boat in still water.
Sharing, area, and formula problems
Other problems lead to rational equations when a quantity is shared among an unknown number of people, or when a formula already contains reciprocals.
Other well-known formulas that involve reciprocals:
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Resistors in parallel: $\dfrac1{R}=\dfrac1{R_1}+\dfrac1{R_2}$.
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Concentration $=\dfrac{\text{amount of solute}}{\text{total volume}}$. If only water evaporates, the amount of salt doesn't change: with $50$ L at $0.04$ kg/L there are $50\cdot0.04=2$ kg of salt, and the concentration after $x$ L evaporate is $\dfrac{2}{50-x}$.
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Two poles of heights $h_1$ and $h_2$ with crossed cables: the height $h$ of the crossing satisfies $\dfrac1h=\dfrac1{h_1}+\dfrac1{h_2}$.
A group of students rents a bus for a field trip for a total price of $24\,000\text{ \text{RD\$}\ }$. At the last minute, $5…
A group of students rents a bus for a field trip for a total price of $24\,000\text{ \text{RD\$}\ }$. At the last minute, $5$ more students sign up for the trip, which means that each of those who were originally going pays $240\text{ \text{RD\$}\ }$ less. Set up a rational equation to calculate how many students were in the original group.
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.
In an electronic circuit, two resistors are connected in parallel. The total equivalent resistance, $R_{eq}$, is calculated using the formula…
In an electronic circuit, two resistors are connected in parallel. The total equivalent resistance, $R_{eq}$, is calculated using the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. It is known that the value of resistance $R_2$ is $5\ \Omega$ greater than that of $R_1$, and that the equivalent resistance of the circuit is $6\ \Omega$. Set up and solve a rational equation to find the value of both resistors.
Change of variable and an extension
Change of variable
If the same expression repeats (or its reciprocal appears), call it $t$ and the equation becomes simpler. At the end, undo the change.
Extension: a parameter
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 $t = \frac{x^2+1}{x}$:
$\frac{x^2+1}{x} + \frac{6x}{x^2+1} = 5$
In a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. Water is to be evaporated from the mixture at a…
In a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. Water is to be evaporated from the mixture at a constant rate to increase its concentration (the amount of salt remains unchanged). If $x\text{ L}$ of water is evaporated, the new concentration must be $0.1\text{ kg/L}$. Set up the corresponding rational equation and calculate the amount of water that must be evaporated.
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.