Absolute Value Equations

Solve the following linear equation with absolute value:

$|3x - 5| = 7$

Isolate the absolute value and find the solutions to the equation:

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

Solve the following equation with fractional coefficients:

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

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 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:

$|5x + 2| = -4$

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

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

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:

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

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:

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

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.

Solve the following rational equation with absolute value:

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

Solve the following rational equation involving an absolute value. Pay special attention to the values that make the denominator zero:

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

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 all the real solutions to the combined equation:

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

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:

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

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.

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.

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 $2\,400$ pesos is proportional to the price itself, modeled by the equation $|P - 2\,400| = \frac{P}{5}$. Calculate the exact prices that satisfy this condition.

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.

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.