Exercises: Modulus equations

Solve the following linear equation with a modulus:

$|3x - 5| = 7$

Isolate the modulus and find the solutions of the equation:

$2|x + 4| - 10 = 0$

Find the values of $x$ that satisfy the following quadratic equation with a modulus:

$|x^2 - 9| = 0$

Solve the equation by considering that the arguments of the moduli are equal or opposite:

$|2x - 1| = |x + 3|$

Analyze and solve the following equation. Justify your answer:

$|5x + 2| = -4$

Find the numbers whose distance to the number $4$ on the real line is exactly $9$. Set up and solve the corresponding modulus equation.

Solve the following equation with fractional coefficients:

$\left| \frac{2x}{3} - 1 \right| = 5$

Find all the real solutions of the equation:

$|x^2 - 5x| = 6$

Solve the following equation and check the validity of the solutions obtained, discarding those that do not make sense:

$|x - 5| = 2x - 1$

Find the solutions of the following equation by making an appropriate change of variable or distinguishing cases according to the sign of $x$:

$x^2 - 4|x| - 5 = 0$

A machine cuts steel cables for a suspension bridge. The length $L$ (in feet) 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.

A particle oscillates around an equilibrium point. Its position $x$ at certain critical instants satisfies the equation $|x^2 - 10| = 3x$. Find the exact positions $x$ where this occurs, making sure to discard invalid solutions.

Solve the following rational equation with a modulus:

$\left| \frac{x+2}{x-1} \right| = 3$

In an economic model, the deviation of the price $P$ of a share from its reference value of $\$120$ is proportional to the price itself, modeled by the equation $|P - 120| = \frac{P}{5}$. Calculate the exact prices that meet this condition.

Find the values of $x$ that satisfy the following equation, checking the solutions in the original equation:

$|x^2 - 4x| = 3x - 6$

During a chemical 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 those that lack physical or mathematical sense in this equation.

Solve the following equation with nested moduli:

$||2x - 1| - 4| = 5$

Find the solutions of the equation by dividing the real line into intervals according to the roots of the arguments of the moduli:

$|x - 2| + |x + 3| = 7$

Determine the value of the parameter $k$ for which the equation $|x^2 - 4x + 3| = k$ has exactly three distinct real solutions.

Solve the following rational equation involving a modulus. Pay special attention to the values that make the denominator zero:

$\left| \frac{x^2 - x - 2}{x^2 + x} \right| = 1$

The profile of the entrance of a modern design tent is modeled by the equation $y = |x - 4| + |x - 8|$, where $x$ and $y$ are in feet. To place a horizontal support beam at a height of $y = 6$ feet, solve the equation $|x - 4| + |x - 8| = 6$ to find the coordinates $x$ of the ends of the support.

Find all the real solutions of the combined equation:

$x^2 - 2|x - 1| = 2$

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 coordinates $x$ where the robot can stop.

Solve the following equation by equating the modulus of a quadratic expression to the modulus of a linear expression:

$|x^2 - 5x + 4| = |2x - 6|$