Equations with absolute value

Solve the following linear equation with absolute value:

$|3x - 5| = 7$

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

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

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:

$|5x + 2| = -4$

Consider solving the equation with two equated absolute values in $\mathbb{R}$:

To solve it, three students propose different approaches:

• Method 1 (by intervals): Determine the signs of $2x - 3$ and $x + 4$ by dividing the real line into three regions using the critical points $x = -4$ and $x = 1.5$, removing the bars in a fixed manner in each section.

• Method 2 (symmetry property): Directly apply the equivalence $|A| = |B| \iff A = B \text{ or } A = -B$.

• Method 3 (squaring both sides): Square both sides to obtain $(2x - 3)^2 = (x + 4)^2$ and solve the resulting quadratic equation.

1. a) Solve the equation applying Method 2 and verify the validity of the solutions obtained.
2. b) Solve the equation applying Method 3, expressing it as a difference of squares $(2x - 3)^2 - (x + 4)^2 = 0$ and factoring without expanding the full polynomials.
3. c) Critically compare the three methods: justify why Method 2 is the most efficient in this case and discuss what disadvantages or risks of unnecessary algebraic calculations Methods 1 and 3 present.

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 the solutions to the following equation by making a suitable substitution or distinguishing cases according to the sign of $x$:

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

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.

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:

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

Consider the equation with two added absolute values:

1. a) Estimate: Interpret geometrically $|x - 3|$ and $|x - 7|$ as the distances from a point $x$ on the real number line to the points $3$ and $7$. What is the sum of both distances if $x$ lies within the segment $[3, 7]$? From that observation, reason why the solutions cannot lie in that interval, estimate how many solutions the equation must have, and symmetrically deduce how far from the endpoints $3$ and $7$ they must be located.
2. b) Check: Solve the equation rigorously by dividing the real number line into three intervals delimited by the critical values $x = 3$ and $x = 7$, removing the absolute value bars according to the sign of each argument in each segment.
3. c) Compare the exact values found in section (b) with the deductions you made in the estimation in section (a).

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 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 $|P - 120| = \frac{P}{5}$. 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 $2|x| - |x - 10| = 5$. Find the exact $x$ coordinates where the robot can stop.