Representing integers on the number line

Look at the following number line and identify the values represented by the letters $A$, $B$ and $C$.

Draw a number line in your notebook and represent the following integers on it:

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

Represent the number $-3$ and its opposite on the number line. How far is each one from the origin?

You can use this number line, in which the origin and the units are marked (each mark is one unit), or draw your own:

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

In a ski resort in Sierra Nevada, the thermometer recorded the following temperatures throughout the morning:

• At 8:00: $-6$ $^\circ$C

• At 10:00: $-2$ $^\circ$C

• At 12:00: $+3$ $^\circ$C

1. a) Represent the three temperatures on a vertical number line (like a thermometer).
2. b) What was the temperature change (distance on the line) between 8:00 and 12:00?

Find the integer that is exactly the same distance from $-7$ and from $+3$ on the number line. In the figure those two numbers are marked. Represent it graphically on the line to check your result and explain how you calculated it.

In a maritime shipping channel, two acoustic warning buoys have been placed at integer positions $-8$ and $+6$ on the number line. A repeater station needs to be anchored at an intermediate integer point $c$ such that both buoys are at exactly the same distance $d$ from this repeater.

1. a) Deduce with reasoning the central position $c$ of the repeater and calculate the value of the distance $d$ to either buoy.
2. b) Write an absolute value equation of the form $|x - c| = d$ whose only solutions on the number line accurately represent the positions of both buoys.
3. c) Solve your equation algebraically, verifying that it gives exactly the points $-8$ and $+6$, and explain why an equation of this type always yields two symmetric solutions with respect to $c$.

Write in your notebook all the integers $x$ that satisfy the condition $-2 < x \leq 3$ and represent them on the number line. Pay close attention to the open and closed endpoints.

You can copy this number line into your notebook:

Write down all the integers $x$ that satisfy the following two conditions simultaneously and represent them on a number line:

• $-5 < x \leq 2$

• The absolute value of $x$ is greater than $1$, that is, $|x| > 1$.

A Met Éireann weather station located in the Wicklow Mountains has an automatic frost alarm programmed. The sensor activates the warning signal if the integer temperature $T$ (in degrees Celsius) falls within the risk range from $-7$ ºC to $+2$ ºC, with the condition that the boundary value $-7$ ºC does activate the alert, whereas if the temperature rises to reach exactly $+2$ ºC, the alert is deactivated as it is considered out of danger.

1. a) Write the activation condition as a double inequality for the variable $T$ (an interval).
2. b) Explain what type of geometric representation (open circle or solid circle) corresponds to each of the two endpoints on the number line and what physical meaning each one has in this context.
3. c) Write the complete list of all integer temperature values that activate the alarm and calculate the distance on the number line between both endpoints using subtraction within an absolute value.

Represent on the number line all the integers whose absolute value is equal to $4$. How many numbers have you marked?

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

Represent the number $-3$ and all the integers whose absolute value is less than it on the number line. Is it possible to complete this task? Justify your answer based on the definition of absolute value and its graphical representation.

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

Given the following expressions, place on the number line the values of $x$ that make them true:

1. a) $|x - 3| = 5$
2. b) $|x - 2| = 4$

Which integer is a solution to both expressions at the same time? Mark it with a different colour on your line.