Exercises: 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 values of $x$ that satisfy the following quadratic equation with absolute value:

$|x^2 - 9| = 0$

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:

$|5x + 2| = -4$

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:

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

Find all the real solutions to the equation:

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

On the real number line, the distance between two numbers $x$ and $a$ is expressed as $|x - a|$. Based on this geometric interpretation:

1. a) Write an absolute value equation of the form $|x - c| = d$ (with $d > 0$) whose solution set is exactly $\{-3, 7\}$. Justify your choice for the numerical values of $c$ (center or midpoint) and $d$ (distance to the endpoints).
2. b) Formally solve the equation you stated in the previous section to verify analytically that it yields exactly those two solutions.

To solve the equation $|2x - 8| = -4$, Cian writes the following procedure in his notebook:

Cian concludes that the equation has two solutions: $x = 2$ and $x = 6$.

1. a) Explain with reasoning the conceptual error Cian makes at the starting point, before performing any algebraic manipulation, referring to the geometric definition of absolute value.
2. b) Substitute the values $x = 2$ and $x = 6$ into the original equation to show the arithmetic contradiction his approach leads to.
3. c) Determine what the true solution set of the equation is.

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 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 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.

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:

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

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.

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

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

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.

In an olive oil cooperative in Kilkenny, extra virgin olive oil is bottled in bottles with a nominal volume of $1\text{ litre}$. During a quality control audit, the filling sensor detects that the actual liquid volume $V$ (in litres) of anomalous batches satisfies the following equality:

1. a) Deduce the mathematical condition that the expression on the right-hand side ($0.5 \times V + 0.15 \ge 0$) must satisfy in order for the equation to have a solution, and interpret what it means regarding the admissible volume of the bottles.
2. b) Solve the equation by separating the two corresponding cases and check whether the candidate solutions obtained satisfy the condition from the previous section.
3. c) Calculate the absolute error of each valid volume relative to the nominal litre ($|V - 1|$) and determine which of the two defective containers exhibits a greater volumetric deviation.

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:

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

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

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

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

Find all the real solutions to 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 $x$ 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:

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

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.

We wish to construct an equation with nested absolute values of the form:

where $a$, $b$, and $c$ are real numbers with $b > 0$ and $c > 0$, such that the following three conditions are simultaneously met:

1. The equation has exactly three distinct real solutions.

2. The smallest solution is $x = -2$.

3. The greatest solution is $x = 8$.

1. a) Solve the equation 'from the outside in' by splitting it into the two branches generated by the outer absolute value. From these two branches, reason out what required relationship must exist between the parameters $b$ and $c$ for the equation to yield exactly three solutions (and not two or four).
2. b) Based on the previous deduction and the given extreme solutions ($x = -2$ and $x = 8$), calculate the exact numerical values of the constants $a$, $b$, and $c$.
3. c) Write the final equation with the parameters obtained and solve it step by step to verify that the three solutions found satisfy all the imposed conditions.