Equations with Absolute Value
A guagua fare costs $\text{RD}\$50$ and the driver tells you the price can vary "by 5 pesos up or down" depending on the route. It doesn't matter whether it's more or less: what matters is the distance from 50. That is absolute value, and an equation with absolute value hides two equations at once: what is inside can be positive or negative. At the end of the page you come back to this fare.
Definition and the basic case $|ax+b|=k$
Solve the following linear equation with absolute value: $|3x - 5| = 7$
Solve the following linear equation with absolute value:
$|3x - 5| = 7$
Find the numbers whose distance to the number $4$ on the number line is exactly $9$. Set up and solve the corresponding absolute value equation.
Find the numbers whose distance to the number $4$ on the number line is exactly $9$. Set up and solve the corresponding absolute value equation.
Solve the following equation with fractional coefficients: $\left| \frac{2x}{3} - 1 \right| = 5$
Solve the following equation with fractional coefficients:
$\left| \frac{2x}{3} - 1 \right| = 5$
No-solution cases and two equal absolute values
If the right side is $0$, there is only one case: $|2x+8|=0$ reduces to $2x+8=0$, that is, $x=-4$. And if you first have to isolate the absolute value, do it before splitting: $2|x+4|-10=0$ becomes $|x+4|=5$.
Isolate the absolute value and find the solutions to the equation: $2|x + 4| - 10 = 0$
Isolate the absolute value and find the solutions to the equation:
$2|x + 4| - 10 = 0$
Solve the equation by considering that the arguments of the absolute values are equal or opposite: $|2x - 1| = |x + 3|$
Solve the equation by considering that the arguments of the absolute values are equal or opposite:
$|2x - 1| = |x + 3|$
Analyse and solve the following equation. Justify your answer: $|5x + 2| = -4$
Analyse and solve the following equation. Justify your answer:
$|5x + 2| = -4$
When the other side contains the unknown
Solve the following equation and check the validity of the solutions obtained, discarding any that do not make sense: $|x - 5| = 2x - 1$
Solve the following equation and check the validity of the solutions obtained, discarding any that do not make sense:
$|x - 5| = 2x - 1$
Solve the following rational equation with absolute value: $\left| \frac{x+2}{x-1} \right| = 3$
Solve the following rational equation with absolute value:
$\left| \frac{x+2}{x-1} \right| = 3$
During a chemistry experiment, the margin of error in measuring the pH of a buffer solution is given by the equation $|2p - 14| = p - 4$, where $p$ is the pH value.…
During a chemistry experiment, the margin of error in measuring the pH of a buffer solution is given by the equation $|2p - 14| = p - 4$, where $p$ is the pH value. Calculate the possible values of $p$ and discard any that lack physical or mathematical meaning in this equation.
Extension: quadratic expressions inside the absolute value
If there is a second-degree polynomial inside, the method is the same: two equations, and each one can give up to two solutions.
Find the values of $x$ that satisfy the following quadratic equation with absolute value: $|x^2 - 9| = 0$
Find the values of $x$ that satisfy the following quadratic equation with absolute value:
$|x^2 - 9| = 0$
Find all the real solutions to the equation: $|x^2 - 5x| = 6$
Find all the real solutions to the equation:
$|x^2 - 5x| = 6$
Find the values of $x$ that satisfy the following equation, checking the solutions in the original equation: $|x^2 - 4x| = 3x - 6$
Find the values of $x$ that satisfy the following equation, checking the solutions in the original equation:
$|x^2 - 4x| = 3x - 6$
Extension: cases by the sign of the variable
In $x^2-4|x|-5=0$ it is not enough to "split into $\pm k$": you apply the definition and solve in each zone.
Find the solutions to the following equation by making a suitable substitution or distinguishing cases according to the sign of $x$: $x^2 - 4|x| - 5 = 0$
Find the solutions to the following equation by making a suitable substitution or distinguishing cases according to the sign of $x$:
$x^2 - 4|x| - 5 = 0$
A particle oscillates around an equilibrium point. Its position $x$ at certain critical moments satisfies the equation $|x^2 - 10| = 3x$. Find the exact positions $x$…
A particle oscillates around an equilibrium point. Its position $x$ at certain critical moments satisfies the equation $|x^2 - 10| = 3x$. Find the exact positions $x$ where this occurs, making sure to discard invalid solutions.
Find all the real solutions to the combined equation: $x^2 - 2|x - 1| = 2$
Find all the real solutions to the combined equation:
$x^2 - 2|x - 1| = 2$
Extension: sums and nested absolute values
If there is one absolute value inside another, you isolate from the outside in. If there are two absolute values added, you divide the line into intervals.
Solve the following equation with nested absolute values: $||2x - 1| - 4| = 5$
Solve the following equation with nested absolute values:
$||2x - 1| - 4| = 5$
Find the solutions to the equation by dividing the number line into intervals according to the roots of the arguments of the absolute values: $|x - 2| + |x + 3| = 7$
Find the solutions to the equation by dividing the number line into intervals according to the roots of the arguments of the absolute values:
$|x - 2| + |x + 3| = 7$
A robotic arm moves along a linear rail. Its distance to two beacons located at positions $x=0$ and $x=10$ must satisfy the constraint given by the equation…
A robotic arm moves along a linear rail. Its distance to two beacons located at positions $x=0$ and $x=10$ must satisfy the constraint given by the equation $2|x| - |x - 10| = 5$. Find the exact $x$ coordinates where the robot can stop.
Solve the following equation by equating the absolute value of a quadratic expression to the absolute value of a linear expression: $|x^2 - 5x + 4| = |2x - 6|$
Solve the following equation by equating the absolute value of a quadratic expression to the absolute value of a linear expression:
$|x^2 - 5x + 4| = |2x - 6|$
Extension: the graph and the solutions depending on a parameter
Solving $|f(x)|=k$ means finding where the graph of $y=|f(x)|$ crosses the horizontal line $y=k$. The graph of $|f|$ is obtained from the graph of $f$ by reflecting upward the part that lies below the $x$-axis. Move $k$ and count the crossings.
Determine the value of the parameter $k$ for which the equation $|x^2 - 4x + 3| = k$ has exactly three distinct real solutions. In the figure you have the graph of…
Determine the value of the parameter $k$ for which the equation $|x^2 - 4x + 3| = k$ has exactly three distinct real solutions.
In the figure you have the graph of $f(x)=|x^2-4x+3|$. Use it to reason how many solutions the equation has depending on the height of the horizontal line $y=k$, and justify the value of $k$ with calculations.
Applications of absolute value
A machine cuts steel cables for a suspension bridge. The length $L$ (in meters) of a cable is considered acceptable if it satisfies the tolerance equation…
A machine cuts steel cables for a suspension bridge. The length $L$ (in meters) of a cable is considered acceptable if it satisfies the tolerance equation $|2L - 150| = 1.5$. Calculate the maximum and minimum acceptable lengths for the cable.
In an economic model, the deviation of the price $P$ of a share from its benchmark value of $7\,200$ pesos is proportional to the price itself, modeled by the equation…
In an economic model, the deviation of the price $P$ of a share from its benchmark value of $7\,200$ pesos is proportional to the price itself, modeled by the equation $|P - 7\,200| = \frac{P}{5}$. Calculate the exact prices that satisfy this condition.
The entrance profile of a modern-design tent is modeled by the equation $y = |x - 4| + |x - 8|$, where $x$ and $y$ are in meters. To place a horizontal support beam at a…
The entrance profile of a modern-design tent is modeled by the equation $y = |x - 4| + |x - 8|$, where $x$ and $y$ are in meters. To place a horizontal support beam at a height of $y = 6$ meters, solve the equation $|x - 4| + |x - 8| = 6$ to find the $x$ coordinates of the ends of the support.