Malik and Emma are mowing lawns for the summer. Together they finish the big lawn at the park in 4 hours. Emma alone would take 6 hours less than Malik alone. How long would Malik take working alone? If Malik needs $x$ hours, Emma needs $x-6$, and the unknown ends up in a denominator. By the end of this page you will know how to set up and solve equations like this one, and how to catch the answers that look right but are not.
Excluded values and the basic method
The plan is always the same: get rid of the fractions by multiplying by the least common denominator (LCD), which is the least common multiple of the denominators. The danger is also always the same: a number that makes a denominator $0$ can never be a solution.
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Solve the rational equation by first finding a common denominator: $\frac{1}{x} + \frac{1}{2x} + \frac{1}{3x} = \frac{11}{6}$
Solve the rational equation by first finding a common denominator:
When a denominator is a polynomial, factor it before you look for the LCD. Then you can see the excluded values directly in the factors, and the LCD does not get bigger than it has to be.
If a denominator is a cubic or higher, find one zero by testing small integers, then use synthetic division to pull out that factor.
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Find the solutions of the following equation: $\frac{4}{x-2} - \frac{3}{x+2} = \frac{1}{x^2-4}$
Why does this happen? When you multiply by an expression like $(x-2)$, you may be multiplying both sides by $0$ at $x=2$. Any equation turns into the true statement $0=0$ there, so a value that was never a solution can sneak in.
A graph shows an extraneous solution too. The graph of $y=\dfrac{x^2-4}{x-2}$ is the line $y=x+2$ with a hole at $x=2$. The line would meet $y=4$ at $x=2$, but that point is a hole, so the equation $\dfrac{x^2-4}{x-2}=4$ has no solution.
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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:
Find the solutions of the following equation, discarding those 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 of the following equation, discarding those that turn out to be false (extraneous solutions):
Often a numerator and a denominator share a factor. You can cancel it and work with a simpler equation. The excluded values still come from the original denominators.
Two special factorizations come up a lot. You can check them by multiplying out.
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Determine all the real solutions of the equation, simplifying the algebraic fractions using algebraic identities before operating:…
Determine all the real solutions of the equation, simplifying the algebraic fractions using algebraic identities before operating:
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 the parameter $a$ the equation…
Sometimes the same expression shows up twice, once as itself and once flipped upside down. Rename it with a new letter and the equation becomes much easier.
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Solve the following rational equation by making the change of variable $t = \frac{x^2+1}{x}$: $\frac{x^2+1}{x} + \frac{6x}{x^2+1} = 5$
Solve the following rational equation by making the change of variable $t = \frac{x^2+1}{x}$:
Work. If a person finishes a job in $t$ hours, their rate is $\dfrac1t$ of the job per hour. When people work together, their rates add. If A needs $a$ hours alone, B needs $b$ hours alone, and together they need $T$ hours:
$$\frac1a+\frac1b=\frac1T$$
Travel. Since distance $=$ rate $\times$ time, we have $\text{time}=\dfrac{\text{distance}}{\text{rate}}$. A boat going with a current of $c$ moves at (still-water speed) $+\,c$, and against it at (still-water speed) $-\,c$.
Back to the start: Malik needs 12 hours alone.
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Two 3D printers working together take $6$ hours to manufacture a complete batch of parts for an engineering project. If they worked separately, printer A would take $5$…
Two 3D printers working together take $6$ hours to manufacture a complete 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 travel 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 entire…
A freight train must travel 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 entire journey. Set up a rational equation and calculate the initial average speed of the train.
A motorboat makes a journey along a river, traveling $24\text{ km}$ downstream and then returning to the starting point upstream. The total navigation time was…
A motorboat makes a journey along a river, traveling $24\text{ km}$ 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.
Three common situations give rational equations. In each one, ask yourself: what stays the same?
Sharing a cost. The total stays the same, so cost per person $=\dfrac{\text{total}}{\text{number of people}}$.
Area. The area stays the same, so height $=\dfrac{\text{area}}{\text{base}}$.
Mixtures. The amount of the dissolved substance stays the same (for example when water evaporates), so concentration $=\dfrac{\text{amount of substance}}{\text{total amount}}$.
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A group of students rents a bus for a trip for a total price of $\$400\text{ \$}$. At the last minute, $5$ more students join the trip, which means that each of the…
A group of students rents a bus for a trip for a total price of $\$400\text{ \$}$. At the last minute, $5$ more students join the trip, which means that each of the original students pays $\$4\text{ \$}$ less. Set up a rational equation to calculate how many students made up the original group.
A graphic designer is creating a rectangular poster whose area must be exactly $600\text{ cm}^2$. If the base of the poster were increased by $10\text{ cm}$ and the…
A graphic designer is creating a rectangular poster whose area must be exactly $600\text{ cm}^2$. If the base of the poster were increased by $10\text{ cm}$ and the height reduced 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 a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. It is desired to evaporate water from the mixture at a…
In a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. It is desired to evaporate water 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.
Several different situations share the same equation shape:
$$\frac1a+\frac1b=\frac1c$$
Work problems use it, and so do two resistors connected in parallel ($\tfrac{1}{R_{eq}}=\tfrac1{R_1}+\tfrac1{R_2}$) and two crossing cables or ladders. Whatever the story, the strategy is the same: write each unknown in terms of one variable, then multiply by the LCD.
For crossing posts of heights $h_1$ and $h_2$, the crossing height $h$ satisfies $\dfrac1h=\dfrac1{h_1}+\dfrac1{h_2}$.
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In an electronic circuit, two resistors are connected in parallel. The equivalent resistance of the combination, $R_{eq}$, is calculated using the formula…
In an electronic circuit, two resistors are connected in parallel. The equivalent resistance of the combination, $R_{eq}$, is calculated using the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. It is known that the value of resistor $R_2$ 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.
Two vertical posts of heights $x$ and $x+2$ feet are separated by a certain distance on flat ground. Two straight, taut cables are stretched from the top of each post to…
Two vertical posts of heights $x$ and $x+2$ feet are separated by a certain distance on flat ground. Two straight, taut cables are stretched from the top of each post to the base of the opposite post. The point where the cables cross is at a height of $2.4\text{ ft}$ from the ground. Knowing that the height of the crossing $h$ satisfies the geometric relationship $\frac{1}{h} = \frac{1}{h_1} + \frac{1}{h_2}$ (where $h_1$ and $h_2$ are the heights of the posts), set up a rational equation and calculate the height of both posts.