Modulus of a real number

Calculate the numerical value of the following expressions using the definition of modulus:

1. a) $|-12.5|$

1. b) $|7 - 15|$

1. c) $|\pi - 4|$

1. d) $|1 - \sqrt{2}|$

The temperature in a town in the Irish Midlands at 6:00 a.m. was $-3\,^\circ C$, while at 3:00 p.m. it rose to $14\,^\circ C$. Use the concept of absolute value to express and calculate the temperature distance (the difference in degrees) between the two measurements.

In the figure you can see the two temperatures placed on the number line of degrees.

1. a) Write, using absolute value, the expression that gives the distance between the two temperatures.

1. b) Calculate its value and interpret it in the context of the problem.

Define the following absolute value function as a piecewise function:

State the expression of each branch and the interval on which each one is valid.

Find the values of $x$ that satisfy the following equalities:

1. a) $|x| = 8$

1. b) $|x - 3| = 5$

1. c) $|x + 2| = 0$

Calculate the values of $x$ that satisfy the following equation:

Solve the following equation with nested absolute values and represent the solutions on the real number line:

You can use the following number line as support, on which you must mark the solutions you obtain:

Represent on the real number line and write in interval form the values of $x$ that satisfy the following inequality:

You can do it on paper and send a photo with your answer.

Solve the following absolute value inequality and represent the solution on the real number line. Express the final result using interval notation:

Solve the inequality $|x| > 5$ and represent the solution graphically. Remember that this type of expression gives rise to a union of two infinite rays.

En una fábrica de componentes electrónicos, una resistencia debe tener un valor nominal de $220\ \Omega$. Por control de calidad, se acepta que el valor real $x$ de la resistencia varíe respecto al nominal en un máximo de $5\ \Omega$.

En la recta de la figura se marca el valor nominal y un valor real cualquiera $x$, junto con su distancia al valor nominal.

1. a) Escribe esta condición utilizando una desigualdad con valor absoluto.

1. b) Resuelve la desigualdad y determina el intervalo de valores óhmicos aceptables para estas resistencias.

In a statistical study on the monthly leisure expenditure of young people in a city, it has been determined that the mean is $45$ € with a maximum error of $12$ €.

1. a) Express this situation using a neighbourhood $E(a, r)$.
2. b) Write the corresponding modulus inequality.
3. c) Could a young person spending $32$ € per month be included in this study group? Justify your answer.

In a quality control check at an electronics components company, a $1500 \Omega$ resistor is accepted as valid if its actual value $R$ does not differ from the nominal value by more than $2\%$.

1. a) Write an inequality using modulus that represents the acceptable values of $R$.
2. b) Determine the interval of values of $R$ that would cause the component to be rejected due to being above the limit.

Find the interval resulting from the following combined condition and represent it graphically:

A financial analyst observes that the price of a cryptocurrency has fluctuated following the rule $|P - 450| < 75$, where $P$ is the price in euros. At the same time, another source indicates that for security reasons, the investment is only recommended if $|P - 500| > 50$.

Determine the price interval in which both conditions are satisfied to make a safe investment according to these data.

Determine the set of real numbers $x$ that satisfy the following nested absolute value condition:

1. a) Solve the inequality and express the result using interval notation.

1. b) Represent the solution set on the real number line (you can use the number line in the figure, which is left blank, as a guide).

1. c) Describe the solution as a union of (closed) neighborhoods, stating the center and radius of each one.