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 plot the following integers on it:

Plot the number $-3$ and its opposite on the number line. What is the distance from each to the origin?

A thermometer shows a temperature of $4^\circ\text{C}$ at 10:00 a.m. If by 8:00 p.m. the temperature has dropped by $6^\circ\text{C}$, plot both temperatures on a number line and indicate the final temperature.

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

Plot on the number line all the integers whose absolute value is equal to $4$. How many numbers did you mark?

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

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

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

Write all the integers $x$ that simultaneously satisfy the following two conditions 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 starts his shift at "Point Zero" on a main avenue. He makes the following trips:

1. He travels $4$ miles to the east ($+4$).

2. From there, he travels $7$ miles to the west.

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

Draw a number line in your notebook, represent each movement using arrows, and determine at which mile marker he is when he finishes.

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

Given the integers $x$ such that their distance to the origin ($0$) is $6$ units:

1. a) Plot these numbers on the number line.
2. b) Plot their opposites as well.
3. c) What do you observe regarding the position of the original numbers and their opposites on the number line?

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

• At 8:00 a.m.: $-6$ $^\circ$F

• At 10:00 a.m.: $-2$ $^\circ$F

• At 12:00 p.m.: $+3$ $^\circ$F

1. a) Plot the three temperatures on a vertical number line (as if it were a thermometer).
2. b) What was the temperature variation (distance on the line) between 8:00 a.m. and 12:00 p.m.?

Draw a number line and plot 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 $-2$.

Find the integer that is equidistant from $-7$ and $+3$ on the number line. Graph it 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 entrance to the cave. We know that the explorer made the following successive movements:

1. He starts at an unknown point $x$.

2. He moves forward $+12$ units.

3. He moves backward twice his initial position $x$.

4. He moves forward the opposite of $-5$.

If he ends up at point $-3$, represent the situation on the number line and determine the value of the starting point $x$, explaining your reasoning.

On the integer number line, point $A$ represents the number $-4$. Point $B$ is a number such that its distance to the origin is three times the distance from $A$ to the origin. If point $C$ is the midpoint of the line segment formed by the opposite points of $A$ and $B$, what numbers could $C$ represent? Keep in mind that there are two possible solutions for $B$. Represent them on a graph.

Identify all the integers $n$ that simultaneously satisfy the following three conditions and place 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 located on the same vertical line. If we plot their positions on a horizontal number line where $0$ is sea level:

• The submarine is at point $S$.

• The helicopter is at point $H$.

• It is known that $|S| = |H| - 200$.

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

Draw the number line and find the coordinates of $S$ and $H$, knowing that the helicopter is flying 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 that $|R| = |P| + |Q|$.

Find the values of $P$, $Q$, and $R$. Represent all possible positions of $R$ on the number line, and explain which one is farthest from the midpoint between $P$ and $Q$.

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

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 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 twice the absolute value of $b$, that is, $|a| = 2 \cdot |b|$, find the values of $a$ and $b$ and represent them. What is the opposite of the sum $a + b$?

Draw a number line and graph the following sets of numbers so that the difference between them can be seen:

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

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

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