Equations with absolute value
The absolute value $|x|$ measures the distance from $x$ to $0$ on the real line, regardless of sign. An equation with absolute value actually hides two equations at once: the expression inside can be positive or negative, and both cases must be solved and checked. In Bachillerato this is treated as a topic in its own right because it appears in domains, inequalities and, later on, in limits and continuity.
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.
Cases with no solution and equal absolute values
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:
$|5x + 2| = -4$
Absolute value of quadratic expressions
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:
$|x^2 - 5x| = 6$
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:
$|x - 5| = 2x - 1$
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:
$|x^2 - 4x| = 3x - 6$
Nested absolute values and solving by 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:
$|x - 2| + |x + 3| = 7$
Applications of absolute value
A machine cuts steel cables for a suspension bridge. The length $L$ (in metres) 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 metres) 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 $120$ euros is proportional to the price itself, modelled by the equation…
In an economic model, the deviation of the price $P$ of a share from its benchmark value of $120$ euros is proportional to the price itself, modelled by the equation $|P - 120| = \frac{P}{5}$. Calculate the exact prices that satisfy this condition.
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.