Exercises: Modulus inequalities

Solve the inequality $|x - 5| < 3$ and express the solution as an interval.

Find all real numbers $x$ that satisfy the inequality $|2x + 4| \geq 8$.

Express the following statement using an absolute value inequality: "The distance between a number $x$ and $7$ is strictly less than $4$". Then, determine the solution set.

Solve the following modulus inequality: $|3x - 1| \leq 5$.

Determine the solution set of the inequality $|-x + 6| > 2$.

Solve the inequality $\left| \frac{x}{3} - 2 \right| \leq 1$.

Justify whether the following inequalities have a solution in the set of real numbers and, if so, find it:

1. a) $|x + 4| \leq 0$
2. b) $|2x - 5| < -1$

Solve the inequality $5 - |x - 2| > 1$.

A transport company for thermolabile medicines in Spain must keep its refrigerated van within the temperature range $[-2\text{ }^\circ\text{C}, 6\text{ }^\circ\text{C}]$ so as not to break the cold chain.

1. a) Write an absolute value inequality of the form $|T - T_0| \leq k$ that exactly models this admissible temperature interval, identifying what $T_0$ (central temperature) and $k$ (tolerance margin) represent geometrically on the real line.
2. b) If due to an occasional thermostat failure the sensor records a temperature of $7,2\text{ }^\circ\text{C}$, check analytically whether the inequality is satisfied and calculate the distance from the optimal temperature $T_0$.

Marcos has to solve the inequality $|3x - 9| \leq -3$ in an exam and submits the following working:

"$-(-3) \leq 3x - 9 \leq -3 \implies 3 \leq 3x - 9 \leq -3$. I add 9 to all sides: $12 \leq 3x \leq 6$. I divide by 3: $4 \leq x \leq 2$. Therefore, the solution is the numbers between 4 and 2".

1. a) Point out two serious conceptual mistakes that Marcos makes: one in his initial interpretation of the absolute value with respect to negative numbers and another in the meaning of the resulting chain of inequalities.
2. b) Without performing any additional algebraic calculation, deduce what the true solution set of the inequality is and justify your answer based on the definition of absolute value as distance on the real line.

In an electronic components factory, the ideal length of a chip is $12,5$ mm. For quality control reasons, a maximum margin of error of $0,15$ mm is accepted. Write a modulus inequality that models this situation for the length $L$ of the chip and solve it to find the interval of acceptable lengths.

Solve the quadratic modulus inequality $|x^2 - 5| < 4$.

Find the values of $x$ that satisfy the inequality $|2x - 3| < x + 1$.

Solve the following system of linear modulus inequalities:

The pH of a swimming pool must be maintained at an optimal level of $7,4$. For the water to be suitable for swimming, the absolute difference between the current pH and the optimal pH must not exceed $5\%$ of the optimal value. Set up the corresponding modulus inequality and determine the range of permitted pH levels.

Solve the rational modulus inequality $\frac{2}{|x - 3|} \geq 1$.

Find the solution set of the inequality $|x - 2| < |x + 4|$ by geometrically interpreting the modulus as the distance between points on the number line.

The temperature $T$ (in degrees Celsius) of an industrial furnace fluctuates during operation such that $|2T - 300| \leq 50$. If for safety reasons in the smelting process the temperature cannot drop below $130^\circ$C at any time, verify mathematically whether the furnace meets the safety regulations guaranteed by its fluctuation.

According to agronomic data from AEMET applied to the province of Jaén, the optimal flowering of olive trees during spring requires the daily average temperature $T$ (in $^\circ\text{C}$) to satisfy $|T - 18| \leq 4$. At the same time, technicians warn that an endemic pest proliferates rapidly only if the ambient temperature satisfies the condition $|2T - 30| \geq 12$.

1. a) Solve both inequalities analytically and express their solutions as intervals.
2. b) Determine the temperature interval in which the olive tree is in optimal flowering conditions and, at the same time, there is a risk of pest development.
3. c) If during an atypical week in April a constant average temperature of $13,5\text{ }^\circ\text{C}$ is recorded, reason whether the farmer should worry about flowering, about the pest, or about neither of the two.

Consider the absolute value inequality $|2x - 5| \leq x + 1$.

1. a) Estimate: Before applying algebraic techniques, analyze the sign of the right-hand side ($x + 1$): what minimum condition must $x$ satisfy for the inequality to have a solution? From that bound and by mentally evaluating some simple integers ($x = 0, 1, 2, 3, 5, 7$), estimate between which two integers the interval of solutions will lie.
2. b) Calculate: Solve the inequality by transforming it into the corresponding double inequality and write the exact solution interval.
3. c) Check: Compare the interval obtained with your initial estimate. Are the integers you anticipated inside? Also check what happens numerically at both endpoints of the interval.

Solve the inequality $|x^2 - 4x| > 3x$. Express the solution as a union of intervals.

Determine for which values of the real parameter $k$ the solution set of the inequality $|x - k| \leq 3$ is strictly contained in the interval $[-1, 8]$.

Solve the following system of combined inequalities:

Find the solution set of the inequality $|x - 1| + |x - 3| \leq 6$.

The figure shows the graph of $f(x) = |x-1| + |x-3|$ (in blue) and the line $y = 6$ (in green). You can use it to check the result you obtain algebraically.

The speed of a vehicle at an instant $t$ (in seconds) is given by the expression $v(t) = |3t - 15|$ m/s. Determine during which time interval the speed of the vehicle is strictly less than $6$ m/s and calculate the total distance travelled in that interval, considering that the vehicle moves in a straight line.

Solve the rational modulus inequality $\left| \frac{2x - 1}{x + 2} \right| \geq 1$.

Consider the inequality $|x^2 - a^2| \leq a^2$, where $a > 0$ is a constant real parameter. Express the solution set of this inequality in terms of the parameter $a$.

In a study of optics and refraction, the deviation of a light beam from its ideal path is given by the function $d(x) = |x| - |x - 2|$. Solve the inequality $|d(x)| \leq 1$ to find the values of $x$ for which the deviation remains within the margins tolerable by the instrument.

To solve the inequality $|x^2 - 2x| \leq |x|$, two students propose different approaches:

• Alba: Squares both sides taking advantage of the fact that both are non-negative, $(x^2 - 2x)^2 \leq x^2$, and solves the resulting polynomial inequality by moving everything to the first side and factoring by difference of squares.

• Bruno: Applies the property $|a \cdot b| = |a| \cdot |b|$ on the left-hand side to factor out $|x|$, analyzes what happens if $x = 0$ separately and, assuming $x \neq 0$, divides both sides by $|x|$ to solve a much simpler linear inequality with absolute value.

1. a) Fully develop both methods to find the solution set.
2. b) Compare both procedures: which one do you consider more efficient and less prone to calculation errors? Justify your answer.
3. c) Explain why Alba's method of squaring is mathematically valid in this inequality, but would be invalid or generate extraneous solutions if the initial inequality had been $|x^2 - 2x| \leq x$.

Consider the family of inequalities depending on the real parameter $m$ given by:

1. a) Based on the geometric interpretation of the absolute value as distance between points on the real line, explain the meaning of the sum $|x - 3| + |x + 1|$ and demonstrate with reasoning that its minimum value for any $x \in \mathbb{R}$ is 4.
2. b) From the previous reasoning, determine with justification for which values of $m$ the inequality:

1. Has no real solution.

2. Has a single point as its solution (if you consider that this is impossible for any $m$, justify why).

3. Has exactly the interval $[-1, 3]$ as its solution.

1. c) If we set $m = 8$, deduce the length and endpoints of the solution interval relying exclusively on symmetry with respect to the midpoint of the segment $[-1, 3]$ and on distances, without setting up sign tables or splitting into three intervals.