Question

lesson 23 | session 2 name practice locating positive and negative numbers study how the example shows locating positive and negative numbers on a number line. then solve problems 1–6. example plot each rational number as a point on the number line. $1\frac{1}{2}$ $-1\frac{1}{4}$ $-\frac{1}{2}$ the positive number $1\frac{1}{2}$ is located $1\frac{1}{2}$ units to the right of 0. the negative number $-\frac{1}{2}$ is located $\frac{1}{2}$ unit to the left of 0. the negative number $-1\frac{1}{4}$ is located $1\frac{1}{4}$ units to the left of 0. 1 draw a horizontal number line in the space below. plot and label a point for the opposite of each number in the example. 2 a. what is the opposite of $-26$? how many units away from 0 is its location on a horizontal number line, and in which direction? b. what is the opposite of 26? how many units away from 0 is its location on a horizontal number line, and in which direction? c. what number is the opposite of the opposite of $-26$? explain.

Explanation:

Part 2a

Step1: Find the opposite of -26

The opposite of a number \( n \) is \( -n \). So for \( n = -26 \), the opposite is \( -(-26)=26 \).

Step2: Determine distance and direction

The distance of a number from 0 on the number line is its absolute value. \( |26| = 26 \), so it is 26 units away. Since 26 is positive, it is to the right of 0.

Step1: Find the opposite of 26

Using the definition of opposite, the opposite of 26 is \( -26 \).

Step2: Determine distance and direction

The absolute value of -26 is \( | - 26|=26 \), so it is 26 units away. Since -26 is negative, it is to the left of 0.

Step1: Find the opposite of -26

First, the opposite of -26 is 26 (as in part 2a).

Step2: Find the opposite of 26

Then, the opposite of 26 is -26? Wait, no. Wait, let's re - do. Let's denote \( x=-26 \). The first opposite: \( -x = -(-26)=26 \). The second opposite: \( - ( - x)=x=-26 \)? Wait, no, wait the question is "the opposite of the opposite of -26". Let's use the property: the opposite of the opposite of a number \( a \) is \( a \) itself. So for \( a = - 26 \), the opposite of the opposite of \( -26 \) is \( -26 \)? Wait, no, wait: opposite of -26 is 26, opposite of 26 is -26? Wait, no, that can't be. Wait, no, the opposite of a number \( n \) is \( -n \). So opposite of -26 is \( -(-26) = 26 \). Then opposite of 26 is \( - 26 \)? Wait, no, that's a mistake. Wait, no, the property is that the opposite of the opposite of a number is the number itself. So if we have a number \( a \), opposite of \( a \) is \( -a \), opposite of \( -a \) is \( -(-a)=a \). So for \( a=-26 \), opposite of opposite of \( -26 \) is \( -26 \)? Wait, no, \( a=-26 \), opposite of \( a \) is \( -a = 26 \), opposite of \( -a \) is \( -(-a)=a=-26 \). Wait, but that seems contradictory. Wait, no, let's take an example. Let \( a = 5 \), opposite of 5 is -5, opposite of -5 is 5, which is \( a \). So for \( a=-26 \), opposite of opposite of \( a \) is \( a \), which is -26? Wait, no, \( a=-26 \), opposite of \( a \) is \( -a = 26 \), opposite of \( -a \) is \( -(-a)=a=-26 \). So the opposite of the opposite of -26 is -26? Wait, but that seems like we are going back. Wait, no, maybe I messed up the sign. Wait, let's do it step by step. First number: -26. First opposite: change the sign, so 26. Second opposite: change the sign of 26, so -26? Wait, no, that's not right. Wait, no, the opposite of a number is the number that is the same distance from 0 but on the opposite side. So -26 is 26 units to the left of 0. Its opposite is 26 units to the right of 0 (which is 26). Then the opposite of 26 is 26 units to the left of 0 (which is -26). Wait, but that would mean the opposite of the opposite of -26 is -26. But that contradicts the property I thought of earlier. Wait, no, my earlier property was wrong. Wait, no, let's take \( a = 3 \). Opposite of 3 is -3, opposite of -3 is 3, which is \( a \). So for \( a=-26 \), opposite of \( a \) is \( -a = 26 \), opposite of \( -a \) is \( -(-a)=a=-26 \). So in this case, the opposite of the opposite of -26 is -26? Wait, but that seems like we are back to the original number. Wait, yes, because applying the opposite operation twice cancels out. So if you take the opposite of a number, then take the opposite of that result, you get back the original number. So for \( a=-26 \), opposite of \( a \) is \( -a \), opposite of \( -a \) is \( -(-a)=a \). So the opposite of the opposite of -26 is -26.

Step2: Explain

By the property of opposites, if we have a number \( n \), the opposite of \( n \) is \( -n \), and the opposite of \( -n \) is \( -(-n)=n \). So for \( n = - 26 \), the opposite of the opposite of \( -26 \) is \( -26 \) itself.

Answer:

The opposite of -26 is 26. It is 26 units away from 0, to the right.

Part 2b