Modulus inequalities

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

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

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

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

Solve the following system of linear modulus inequalities:

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

Cian 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 Cian 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.

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.

A transport company for thermolabile medicines in Ireland 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 $T_0$.

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

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

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.