Absolute value 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 $\left| \frac{x}{3} - 2 \right| \leq 1$.

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

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) $|2x - 5| < -1$

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.

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.

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.

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

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

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.

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.

Solve the following system of linear modulus inequalities:

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:

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.