example 4
write an equation involving absolute value for each graph.
24.
\begin{tikzpicture}
\draw- (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-2,0) circle (2pt);
\filldrawblue (4,0) circle (2pt);
\end{tikzpicture}
25.
\begin{tikzpicture}
\draw- (-10,0) -- (10,0);
\foreach \x in {-10,-8,-6,-4,-2,0,2,4,6,8,10} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-6,0) circle (2pt);
\filldrawblue (6,0) circle (2pt);
\end{tikzpicture}
26.
\begin{tikzpicture}
\draw- (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-3,0) circle (2pt);
\filldrawblue (5,0) circle (2pt);
\end{tikzpicture}
27.
\begin{tikzpicture}
\draw- (-7,0) -- (3,0);
\foreach \x in {-7,-6,-5,-4,-3,-2,-1,0,1,2,3} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-6,0) circle (2pt);
\filldrawblue (2,0) circle (2pt);
\end{tikzpicture}
28.
\begin{tikzpicture}
\draw- (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-1,0) circle (2pt);
\filldrawblue (1,0) circle (2pt);
\end{tikzpicture}
29.
\begin{tikzpicture}
\draw- (-7,0) -- (3,0);
\foreach \x in {-7,-6,-5,-4,-3,-2,-1,0,1,2,3} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-5,0) circle (2pt);
\filldrawblue (-1,0) circle (2pt);
\end{tikzpicture}
30.
\begin{tikzpicture}
\draw- (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (3,0) circle (2pt);
\filldrawblue (5,0) circle (2pt);
\end{tikzpicture}
31.
\begin{tikzpicture}
\draw- (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-4,0) circle (2pt);
\filldrawblue (4,0) circle (2pt);
\end{tikzpicture}

Question

example 4
write an equation involving absolute value for each graph.
24.
\begin{tikzpicture}
\draw-> (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-2,0) circle (2pt);
\filldrawblue (4,0) circle (2pt);
\end{tikzpicture}
25.
\begin{tikzpicture}
\draw-> (-10,0) -- (10,0);
\foreach \x in {-10,-8,-6,-4,-2,0,2,4,6,8,10} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-6,0) circle (2pt);
\filldrawblue (6,0) circle (2pt);
\end{tikzpicture}
26.
\begin{tikzpicture}
\draw-> (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-3,0) circle (2pt);
\filldrawblue (5,0) circle (2pt);
\end{tikzpicture}
27.
\begin{tikzpicture}
\draw-> (-7,0) -- (3,0);
\foreach \x in {-7,-6,-5,-4,-3,-2,-1,0,1,2,3} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-6,0) circle (2pt);
\filldrawblue (2,0) circle (2pt);
\end{tikzpicture}
28.
\begin{tikzpicture}
\draw-> (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-1,0) circle (2pt);
\filldrawblue (1,0) circle (2pt);
\end{tikzpicture}
29.
\begin{tikzpicture}
\draw-> (-7,0) -- (3,0);
\foreach \x in {-7,-6,-5,-4,-3,-2,-1,0,1,2,3} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-5,0) circle (2pt);
\filldrawblue (-1,0) circle (2pt);
\end{tikzpicture}
30.
\begin{tikzpicture}
\draw-> (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (3,0) circle (2pt);
\filldrawblue (5,0) circle (2pt);
\end{tikzpicture}
31.
\begin{tikzpicture}
\draw-> (-5,0) -- (5,0);
\foreach \x in {-5,-4,-3,-2,-1,0,1,2,3,4,5} \draw (\x,0.1) -- (\x,-0.1) nodebelow {\x};
\filldrawblue (-4,0) circle (2pt);
\filldrawblue (4,0) circle (2pt);
\end{tikzpicture}

Accepted Answer

Let's solve problem 24 as an example (we can solve others similarly).

### Problem 24:
#### Step 1: Find the midpoint (average) of the two points.
The two points are at $ x = -2 $ and $ x = 4 $. The midpoint $ h $ is calculated as:
$$
h=\frac{-2 + 4}{2}=\frac{2}{2}=1
$$
#### Step 2: Find the distance from the midpoint to either point (this is the radius $ r $).
Distance from $ 1 $ to $ 4 $ (or to $ -2 $):
$$
r=\vert4 - 1\vert=\vert-2 - 1\vert = 3
$$
#### Step 3: Write the absolute - value equation.
The general form of an absolute - value equation for a number line graph with solutions $ x_1 $ and $ x_2 $ is $ \vert x - h\vert=r $, where $ h $ is the midpoint and $ r $ is the distance from the midpoint to each solution.
Substituting $ h = 1 $ and $ r = 3 $ into the formula, we get:
$$
\vert x - 1\vert=3
$$

### Problem 25:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-6 $ and $ x = 6 $. The midpoint $ h $ is:
$$
h=\frac{-6 + 6}{2}=0
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ 0 $ to $ 6 $ (or to $ -6 $):
$$
r=\vert6-0\vert=\vert - 6-0\vert=6
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h = 0 $ and $ r = 6 $, we have:
$$
\vert x\vert=6
$$

### Problem 26:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-3 $ and $ x = 5 $. The midpoint $ h $ is:
$$
h=\frac{-3 + 5}{2}=\frac{2}{2}=1
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ 1 $ to $ 5 $ (or to $ -3 $):
$$
r=\vert5 - 1\vert=\vert-3 - 1\vert = 4
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h = 1 $ and $ r = 4 $, we get:
$$
\vert x - 1\vert=4
$$

### Problem 27:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-6 $ and $ x = 2 $. The midpoint $ h $ is:
$$
h=\frac{-6+2}{2}=\frac{-4}{2}=-2
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ - 2 $ to $ 2 $ (or to $ -6 $):
$$
r=\vert2-(-2)\vert=\vert-6 - (-2)\vert=\vert4\vert = 4
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h=-2 $ and $ r = 4 $, we have:
$$
\vert x+2\vert=4
$$

### Problem 28:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-1 $ and $ x = 1 $. The midpoint $ h $ is:
$$
h=\frac{-1 + 1}{2}=0
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ 0 $ to $ 1 $ (or to $ -1 $):
$$
r=\vert1-0\vert=\vert-1 - 0\vert = 1
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h = 0 $ and $ r = 1 $, we get:
$$
\vert x\vert=1
$$

### Problem 29:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-5 $ and $ x=-1 $. The midpoint $ h $ is:
$$
h=\frac{-5+(-1)}{2}=\frac{-6}{2}=-3
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ -3 $ to $ -1 $ (or to $ -5 $):
$$
r=\vert-1-(-3)\vert=\vert-5 - (-3)\vert=\vert2\vert = 2
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h=-3 $ and $ r = 2 $, we have:
$$
\vert x + 3\vert=2
$$

### Problem 30:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x = 3 $ and $ x = 5 $. The midpoint $ h $ is:
$$
h=\frac{3 + 5}{2}=\frac{8}{2}=4
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ 4 $ to $ 5 $ (or to $ 3 $):
$$
r=\vert5 - 4\vert=\vert3 - 4\vert = 1
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h = 4 $ and $ r = 1 $, we get:
$$
\vert x - 4\vert=1
$$

### Problem 31:
#### Step 1: Find the midpoint of the two points.
The two points are at $ x=-4 $ and $ x = 4 $. The midpoint $ h $ is:
$$
h=\frac{-4 + 4}{2}=0
$$
#### Step 2: Find the distance from the midpoint to either point.
Distance from $ 0 $ to $ 4 $ (or to $ -4 $):
$$
r=\vert4-0\vert=\vert-4 - 0\vert = 4
$$
#### Step 3: Write the absolute - value equation.
Using the formula $ \vert x - h\vert=r $ with $ h = 0 $ and $ r = 4 $, we have:
$$
\vert x\vert=4
$$

### Answers:
24. $\boldsymbol{\vert x - 1\vert=3}$

25. $\boldsymbol{\vert x\vert=6}$

26. $\boldsymbol{\vert x - 1\vert=4}$

27. $\boldsymbol{\vert x + 2\vert=4}$

28. $\boldsymbol{\vert x\vert=1}$

29. $\boldsymbol{\vert x + 3\vert=2}$

30. $\boldsymbol{\vert x - 4\vert=1}$

31. $\boldsymbol{\vert x\vert=4}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 24:

Step 1: Find the midpoint (average) of the two points.

Step 2: Find the distance from the midpoint to either point (this is the radius \( r \)).

Step 3: Write the absolute - value equation.

Problem 25:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either point.

Step 3: Write the absolute - value equation.

Problem 26:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either point.

Step 3: Write the absolute - value equation.

Problem 27:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either point.

Step 3: Write the absolute - value equation.

Problem 28:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either point.

Step 3: Write the absolute - value equation.

Problem 29:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either point.

Step 3: Write the absolute - value equation.

Problem 30:

Step 1: Find the midpoint of the two points.

Step 2: Find the distance from the midpoint to either…

Answer: