Exercises: Representation 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.

A thermometer shows a temperature of $4^\circ\text{C}$ at ten in the morning. If at eight in the evening the temperature has dropped by $6^\circ\text{C}$, represent both temperatures on a line and indicate the final temperature.

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:

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.

A scuba diver is at a depth of $8$ meters below sea level. If they decide to go up $3$ meters to look at some fish, what depth are they at now? Represent the movement using arrows on a horizontal number line.

Place the following numbers on the number line and use it to order them from least to greatest using the symbol $<$:

On the following number line, four points represented by letters have been marked. Identify which integer corresponds to each of the letters $A$, $B$, $C$, and $D$.

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 food delivery driver begins his shift at "Point Zero" on a main avenue. He makes the following trips:

1. Travels $4$ kilometers to the east ($+4$).

2. From there, travels $7$ kilometers to the west.

3. Finally, travels another $2$ kilometers to the west to deliver his last order.

Draw a number line in your notebook, represent each movement using arrows, and determine which kilometer mark he is at when finished.

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 integers $x$ whose distance from the origin ($0$) is $6$ units, and using the number line in the figure (or drawing an identical one in your notebook):

1. a) Represent those numbers on the number line.

1. b) Also represent their opposites.

1. c) What do you notice about the position of the original numbers and their opposites on the line?

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?

Draw a number line and place the following elements on it. Then, order them from least to greatest using the symbol $<$:

• The number $-4$.

• The opposite of $3$.

• The absolute value of $-5$.

• The number $0$.

• The opposite of the opposite of $-2$.

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

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.

An explorer is in a network of underground caves represented on a horizontal number line. The origin $0$ is the cave entrance. We know that the explorer has made the following successive movements:

1. Starts at an unknown point $x$.

2. Moves forward $+12$ units.

3. Moves back twice their initial position $x$ (that is, $2x$ units to the left).

4. Moves forward the opposite of $-5$.

If they finally end up at point $-3$, represent the situation on the number line (draw the line and the movements yourself) and determine, with reasoning, the value of the starting point $x$.

On the number line of integers, point $A$ represents the number $-4$. Point $B$ is a number such that its distance from the origin is three times the distance of $A$ from the origin. If point $C$ is the midpoint of the line segment formed by the opposite points of $A$ and $B$, which numbers could $C$ represent? Note that there are two possible solutions for $B$. Represent them graphically.

Identify all the integers $n$ that satisfy the following three conditions simultaneously and plot them on a number line:

1. The absolute value of $n$ is less than or equal to $7$: $|n| \leq 7$.

2. $n$ is strictly greater than its opposite: $n > -n$.

3. The distance between $n$ and the number $-2$ is greater than $5$ units.

A submarine and a rescue helicopter are positioned on the same vertical line. If we transpose their positions to a horizontal number line where $0$ is sea level:

• The submarine is located at point $S$.

• The helicopter is located at point $H$.

• It is known that $|S| = |H| - 200$.

• The distance between the submarine and the helicopter is $500$ meters.

Draw the number line and find the coordinates of $S$ and $H$, knowing that the helicopter flies above sea level and the submarine is submerged.

Three points $P$, $Q$, and $R$ are located on the number line. It is known that:

• Point $Q$ is the opposite of $-6$.

• Point $P$ is at a distance of $10$ units from $Q$, to the left.

• Point $R$ satisfies $|R| = |P| + |Q|$.

Calculate the values of $P$, $Q$, and $R$. Represent all possible positions of $R$ on the line and explain which of them is furthest from the midpoint between $P$ and $Q$.

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

Given the following expressions, plot 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 in a different color on your number line.

Consider a negative integer $a$ and a positive integer $b$. On the number line, the distance between $a$ and $b$ is $15$ units. If the absolute value of $a$ is double the absolute value of $b$, that is, $|a| = 2 \times |b|$, find the values of $a$ and $b$ and represent them. What is the opposite of the sum $a + b$?

Draw a number line and represent the following sets of numbers so that the difference between them can be seen (for example, by using a different color for each set):

• Set $X$: integers $x$ such that $|x| < 4$.

• Set $Y$: integers $y$ such that their opposite $-y$ is between $-2$ and $5$ (both excluded).

State, with reasoning, which integers belong to both sets at the same time.

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