Update Final_Exam_Study_Sheet.md

2025-01-23 16:11:46 -05:00 · 2019-04-18 23:59:38 +00:00 · 2019-04-18 23:59:38 +00:00 · 1c74d2b042
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 ## Scientific Notation
 -  They convey accuracy and precision. It can either be written as its original number or in scientific notation: 
- 555 (**Exact**) or 5.55 x 10<sup>2</sup> (**3 significant figures**).   
- In scientific notation, values are written in the form **a(10<sup>n</sup>)**, where ```a``` is a number within 1 and 10 and ```n``` is any integer. 
- Some examples include the following: 5.4 x 10<sup>3</sup>, 3.0 x 10<sup>2</sup>, and 4.56 x 10<sup>-4</sup>.    
+- 555 (**Exact**) or $`5.55 \times 10^2`$ (**3 significant figures**).   
+- In scientific notation, values are written in the form $`a(10^n)`$, where $`a`$ is a number within 1 and 10 and $`n`$ is any integer. 
+- Some examples include the following: $`5.4 \times 10^3, 3.0 \times 10^2`$, and $`4.56 \times 10^{-4}`$.    
 - When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used    

 - <img src="https://embedwistia-a.akamaihd.net/deliveries/d2de1eb00bafe7ca3a2d00349db23a4117a8f3b8.jpg?image_crop_resized=960x600" width="500">     
@ -340,20 +340,20 @@ x, & \text{if  } x > 0\\
 ## 2D Geometry Equations
 |Shape|Formula|Picture|
 |:----|:------|:------|
- |Rectangle|```Area```: lw <br> ```Perimeter```: 2(l+w)|<img src="https://lh5.googleusercontent.com/Ib1Evz5PUwd4PzRmFkHj9IY2Is-UthHoUyyiEHAzkJP-296jZvMmHJM1Kws4PmuTeYHV2ZBIJenc4W1pKtsSHvU82lyjOed2XKBb1PWnoaeJ3sSPuaJgSTg8JWbxrvplabCanvTD" width="200">|   
- |Triangle|```Area```: bh/2 <br> ```Perimeter```: a+b+c|<img src="https://lh6.googleusercontent.com/covvHwXxQhrK2Hr0YZoivPkHodgstVUpAQcjpg8sIKU25iquSHrRd2EJT64iWLsg_75WnBw4T9P0OTBiZDkpqEkXxflZQrL16sNhcFfet_z4Mw5EPFgdx_4HzsagV0Sm5jN6EKr_" width="200">|
- |Circle|```Area```: πr<sup>2</sup> <br> ```Circumference```: 2πr or πd|<img src="https://lh5.googleusercontent.com/RydffLVrOKuXPDXO0WGPpb93R8Ucm27qaQXuxNy_fdEcLmuGZH4eYc1ILNmLEx8_EYrRuOuxFavtL9DF1lTWYOx9WaYauVlu0o_UR6eZLeGewGjFNUQSK8ie4eTm1BMHfRoQWHob" width="200">|
- |Trapezoid|```Area```: (a+b)h/2 <br> ```Perimeter```: a+b+c+d|<img src="https://lh6.googleusercontent.com/_nceVtFlScBbup6-sPMulUTV3MMKu1nonei0D1WY-KRkpHSbPCIWgDO8UGDQBGKh8i0dkAqOhFUHl7YHCFOt6AMRSJiXALlBBY0mBo1MMZxHRVcg8DknSlv4ng7_QswcZtaRwrJb" width="200">|
+ |Rectangle|```Area```: $`lw`$ <br> ```Perimeter```: $`2(l+w)`$|<img src="https://lh5.googleusercontent.com/Ib1Evz5PUwd4PzRmFkHj9IY2Is-UthHoUyyiEHAzkJP-296jZvMmHJM1Kws4PmuTeYHV2ZBIJenc4W1pKtsSHvU82lyjOed2XKBb1PWnoaeJ3sSPuaJgSTg8JWbxrvplabCanvTD" width="200">|   
+ |Triangle|```Area```: $`\frac{bh}{2}`$ <br> ```Perimeter```: $`a+b+c`$|<img src="https://lh6.googleusercontent.com/covvHwXxQhrK2Hr0YZoivPkHodgstVUpAQcjpg8sIKU25iquSHrRd2EJT64iWLsg_75WnBw4T9P0OTBiZDkpqEkXxflZQrL16sNhcFfet_z4Mw5EPFgdx_4HzsagV0Sm5jN6EKr_" width="200">|
+ |Circle|```Area```: $`πr^2`$ <br> ```Circumference```: $`2πr`$ or $`πd`$|<img src="https://lh5.googleusercontent.com/RydffLVrOKuXPDXO0WGPpb93R8Ucm27qaQXuxNy_fdEcLmuGZH4eYc1ILNmLEx8_EYrRuOuxFavtL9DF1lTWYOx9WaYauVlu0o_UR6eZLeGewGjFNUQSK8ie4eTm1BMHfRoQWHob" width="200">|
+ |Trapezoid|```Area```: $` \frac{(a+b)h}{2}`$ <br> ```Perimeter```: $`a+b+c+d`$|<img src="https://lh6.googleusercontent.com/_nceVtFlScBbup6-sPMulUTV3MMKu1nonei0D1WY-KRkpHSbPCIWgDO8UGDQBGKh8i0dkAqOhFUHl7YHCFOt6AMRSJiXALlBBY0mBo1MMZxHRVcg8DknSlv4ng7_QswcZtaRwrJb" width="200">|

 ## 3D Geometry Equations
 |3D Object|Formula|Picture|
 |:----|:------|:------|
- |Rectangular Prism|```Volume```: lwh <br> ```SA```: 2(lw+lh+wh)|<img src="https://lh6.googleusercontent.com/-mqEJ4AMk3xDPfqH5kdVukhtCGl3fgTy2ojyAArla54c7UoAnqKW_bsYSaFySXLplE59pqLIg5ANZAL1f6UEejsrKJwQCfyO7gwUQmSDoJJtQG_WkfHcOFDjidXV4Y4jfU2iA5b-" width="200">|
- |Square Based Pyramid|```Volume```: <sup>1</sup>&frasl;<sub>3</sub>b<sup>2</sup>h <br> ```SA```: 2bs+b<sup>2</sup>|<img src="https://lh5.googleusercontent.com/iqaaJtx-Kx4vFT3Yp6YLOmpDFL7_qk2uh0Z21pgPJMDRgchiUBcHeTWkMrR9mFjxCj8w7za1xwN9bo4UFACPZRMSl-V67uPv9FvDyNJVjedmeehx5K-iUK9sBhObhNsLJpNItkg0" width="200">|
- |Sphere|```Volume```: <sup>4</sup>&frasl;<sub>3</sub>πr<sup>3</sup> <br> ```SA```: 4πr<sup>2</sup>|<img src="https://lh6.googleusercontent.com/DL6ViJLbap2dcSAlZnYKR7c33033g8WuJVvqz0KpzCyIJ0wXyrh5ejoLhrTxlX9uASQlxPmihm8doU1sNbaQxqBcTaPnP-lC6LUrPqzPNi11AHiHQAu3ag7uIGcwzkdC9e5uo1en" width="200">|
- |Cone|```Volume```: <sup>1</sup>&frasl;<sub>3</sub>πr<sup>2</sup>h <br> ```SA```: πrs+πr<sup>2</sup>|<img src="https://lh5.googleusercontent.com/V3iZzX8ARcipdJiPPFYso_il3v_tcrYHZlFnq3qkekRSVBVcj8OzWxMuBqN45aHbv6y-fDH4uY11Gus3KMrvf_Z_TvsfJCwZZ19Ezf7Yj6DzVirp-Gx3V0Qy793ooUwTDmdKW_xq" width="200">|
- |Cylinder|```Volume```: πr<sup>2</sup>h <br> ```SA```: 2πr<sup>2</sup>+2πh|<img src="https://lh5.googleusercontent.com/4uWukD3oNUYBG-fLX2-g58X8at0h74al7BJI5l78LZ0Bu9nXuZnt9dp9xiETeLTqykP-WWFdO_H5by4RkgDVxSENZgootSrAsOUoY2RWubflNOAau1bVFgm9YIe59SmiFlyxwgDV" width="200">|
- |Triangular Prism|```Volume```: ah+bh+ch+bl <br> ```SA```: <sup>1</sup>&frasl;<sub>2</sub>blh|<img src="https://lh3.googleusercontent.com/_oRUVgfdksfUXGKQk3AtrtY70E8jEq-RRK-lB9yKc_Rtio2f2utGAY-rI4UqjWEeTzUoN_r7EiqdZZeeE12EY-fiV55QQKdvnv4y4VaxQ9xt9Izugp6Ox_LqIUpQzPKVldptgKWm" width="200">|
+ |Rectangular Prism|```Volume```: $`lwh`$ <br> ```SA```: $`2(lw+lh+wh)`$|<img src="https://lh6.googleusercontent.com/-mqEJ4AMk3xDPfqH5kdVukhtCGl3fgTy2ojyAArla54c7UoAnqKW_bsYSaFySXLplE59pqLIg5ANZAL1f6UEejsrKJwQCfyO7gwUQmSDoJJtQG_WkfHcOFDjidXV4Y4jfU2iA5b-" width="200">|
+ |Square Based Pyramid|```Volume```: $`\frac{1}{3} b^2 h`$ <br> ```SA```: $`2bs+b^2`$|<img src="https://lh5.googleusercontent.com/iqaaJtx-Kx4vFT3Yp6YLOmpDFL7_qk2uh0Z21pgPJMDRgchiUBcHeTWkMrR9mFjxCj8w7za1xwN9bo4UFACPZRMSl-V67uPv9FvDyNJVjedmeehx5K-iUK9sBhObhNsLJpNItkg0" width="200">|
+ |Sphere|```Volume```: $`\frac{4}{3}  πr^3`$ <br> ```SA```: $`4πr^2`$|<img src="https://lh6.googleusercontent.com/DL6ViJLbap2dcSAlZnYKR7c33033g8WuJVvqz0KpzCyIJ0wXyrh5ejoLhrTxlX9uASQlxPmihm8doU1sNbaQxqBcTaPnP-lC6LUrPqzPNi11AHiHQAu3ag7uIGcwzkdC9e5uo1en" width="200">|
+ |Cone|```Volume```: $` \frac{1}{3} πr^2  h`$ <br> ```SA```: $`πrs+πr^2`$|<img src="https://lh5.googleusercontent.com/V3iZzX8ARcipdJiPPFYso_il3v_tcrYHZlFnq3qkekRSVBVcj8OzWxMuBqN45aHbv6y-fDH4uY11Gus3KMrvf_Z_TvsfJCwZZ19Ezf7Yj6DzVirp-Gx3V0Qy793ooUwTDmdKW_xq" width="200">|
+ |Cylinder|```Volume```: $`πr^2h`$ <br> ```SA```: $`2πr^2+2πh`$|<img src="https://lh5.googleusercontent.com/4uWukD3oNUYBG-fLX2-g58X8at0h74al7BJI5l78LZ0Bu9nXuZnt9dp9xiETeLTqykP-WWFdO_H5by4RkgDVxSENZgootSrAsOUoY2RWubflNOAau1bVFgm9YIe59SmiFlyxwgDV" width="200">|
+ |Triangular Prism|```Volume```: $`ah+bh+ch+bl`$ <br> ```SA```: $` \frac{1}{2} blh`$|<img src="https://lh3.googleusercontent.com/_oRUVgfdksfUXGKQk3AtrtY70E8jEq-RRK-lB9yKc_Rtio2f2utGAY-rI4UqjWEeTzUoN_r7EiqdZZeeE12EY-fiV55QQKdvnv4y4VaxQ9xt9Izugp6Ox_LqIUpQzPKVldptgKWm" width="200">|


 ## Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)
@ -362,18 +362,18 @@ x, & \text{if  } x > 0\\

 |Shape|Maximum Area|Minimum Perimeter|
 |:----|:-----------|:----------------|
- |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width<br>A = lw<br>A<sub>max</sub> = (w)(w)<br>A<sub>max</sub> = w<sup>2</sup>|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.<br>P = 2(l+w)<br>P<sub>min</sub> = 2(w)(w)<br>P<sub>min</sub> = 2(2w)<br>P<sub>min</sub> = 4w|
- |3-sided rectangle|l = 2w<br>A = lw<br>A<sub>max</sub> = 2w(w)<br>A<sub>max</sub> = 2w<sup>2</sup>|l = 2w<br>P = l+w2<br>P<sub>min</sub> = 2w+2w<br>P<sub>min</sub> = 4w|
+ |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width<br>$`A = lw`$<br>$`A_{max} = (w)(w)`$<br>$`A_{max} = w^2`$|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.<br>$`P = 2(l+w)`$<br>$`P_{min} = 2(w)(w)`$<br>$`P_{min} = 2(2w)`$<br>$`P_{min} = 4w`$|
+ |3-sided rectangle|$`l = 2w`$<br>$`A = lw`$<br>$`A_{max} = 2w(w)`$<br>$`A_{max} = 2w^2`$|$`l = 2w`$<br>$`P = l+2w`$<br>$`P_{min} = 2w+2w`$<br>$`P_{min} = 4w`$|


 ### 3D Objects

 |3D Object|Maximum Volumne|Minimum Surface Area|
 |:--------|:--------------|:-------------------|
- |Cylinder(closed-top)|The cylinder must be similar to a cube where h = 2r<br>V = πr<sup>2</sup>h<br>V<sub>max</sub> = πr<sup>2</sup>(2r)<br>V<sub>max</sub> = 2πr<sup>3</sup>|The cylinder must be similar to a cube where h = 2r<br>SA = 2πr<sup>2</sup>+2πrh<br>SA<sub>min</sub> = 2πr<sup>2</sup>+2πr(2r)<br>SA<sub>min</sub> = 2πr<sup>2</sup>+4πr<sup>2</sup><br>SA<sub>min</sub> = 6πr<sup>2</sup>|
- |Rectangular Prism(closed-top)|The prism must be a cube, <br> where l = w = h<br>V = lwh<br>V<sub>max</sub> = (w)(w)(w)<br>V<sub>max</sub> = w<sup>3</sup>|The prism must be a cube, <br>where l = w = h<br>SA = 2lh+2lw+2wh<br>SA<sub>min</sub> = 2w<sup>2</sup>+2w<sup>2</sup>+2w<sup>2</sup><br>SA<sub>min</sub> = 6w<sup>2</sup>|
- |Cylinder(open-top)|h = r<br>V = πr<sup>2</sup>h<br>V<sub>max</sub> = πr<sup>2</sup>(r)<br>V<sub>max</sub> = πr<sup>3</sup>|h = r<br>SA = πr<sup>2</sup>+2πrh<br>SA<sub>min</sub> = πr<sup>2</sup>+2πr(r)<br>SA<sub>min</sub> = πr<sup>2</sup>+2πr<sup>2</sup><br>SA<sub>min</sub> = 3πr<sup>2</sup>|
- |Square-Based Rectangular Prism(open-top)|h = w/2<br>V = lwh<br>V<sub>max</sub> = (w)(w)(<sup>w</sup>&frasl;<sub>2</sub>)<br>V<sub>max</sub> = <sup>w<sup>3</sup></sup>&frasl;<sub>2</sub>|h = w/2<br>SA = w<sup>2</sup>+4wh<br>SA<sub>min</sub> = w<sup>2</sup>+4w(<sup>w</sup>&frasl;<sub>2</sub>)<br>SA<sub>min</sub> = w<sup>2</sup>+2w<sup>2</sup><br>SA<sub>min</sub> = 3w<sup>2</sup>|
+ |Cylinder(closed-top)|The cylinder must be similar to a cube where $`h = 2r`$<br>$`V = πr^2h`$<br>$`V_{max} = πr^2(2r)`$<br>$`V_{max} = 2πr^3`$|The cylinder must be similar to a cube where $`h = 2r`$<br>$`SA = 2πr^2+2πrh`$<br>$`SA_{min} = 2πr^2+2πr(2r)`$<br>$`SA_{min} = 2πr^2+4πr^2`$<br>$`SA_{min} = 6πr^2`$|
+ |Rectangular Prism(closed-top)|The prism must be a cube, <br> where $`l = w = h`$<br>$`V = lwh`$<br>$`V_{max} = (w)(w)(w)`$<br>$`V_{max} = w^3`$|The prism must be a cube, <br>where $`l = w = h`$<br>$`SA = 2lh+2lw+2wh`$<br>$`SA_{min} = 2w^2+2w^2+2w^2`$<br>$`SA_{min} = 6w^2`$|
+ |Cylinder(open-top)|$`h = r`$<br>$`V = πr^2h`$<br>$`V_{max} = πr^2(r)`$<br>$`V_{max} = πr^3`$|$`h = r`$<br>$`SA = πr^2+2πrh`$<br>$`SA_{min} = πr^2+2πr(r)`$<br>$`SA_{min} = πr^2+2πr^2`$<br>$`SA_{min} = 3πr^2`$|
+ |Square-Based Rectangular Prism(open-top)|$`h = \frac{w}{2}`$<br>$`V = lwh`$<br>$`V_{max} = (w)(w)(\frac{w}{2})`$<br>$`V_{max} = \frac{w^3}{2}`$|$`h = \frac{w}{2}`$<br>$`SA = w^2+4wh`$<br>$`SA_{min} = w^2+4w(\frac{w}{2})`$<br>$`SA_{min} = w^2+2w^2`$<br>$`SA_{min} = 3w^2`$|

 ## Labelling
 - Given any polygons, labelling the vertices must always:   
@ -420,10 +420,10 @@ x, & \text{if  } x > 0\\

 ## Slope and Equation of Line   
 - ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```the rate of change```       
- ```Slope Formula```: **m = (y<sub>2</sub>-y<sub>1</sub>)/(x<sub>2</sub>-x<sub>1</sub>)**    
- ```Standard Form```: **ax + by + c = 0**, a&isin;Z, b&isin;Z, c&isin;Z (must be integers and ```a``` must be positive)   
- ```Y-intercept Form```: **y = mx + b**    
- ```Point-slope Form```: **y<sub>2</sub>-y<sub>1</sub> = m(x<sub>2</sub>-x<sub>1</sub>)**   
+- ```Slope Formula```: $`m = \frac{y_2 - y_1}{x_2 - x_1}`$    
+- ```Standard Form```: $`ax + by + c = 0, a \isin \mathbb{Z}, b \isin \mathbb{Z}, c \isin \mathbb{Z}`$ (must be integers and $`a`$ must be positive)   
+- ```Y-intercept Form```: $`y = mx + b`$    
+- ```Point-slope Form```: $`y_2-y_1 = m(x_2-x_1)`$   
 - The slope of a vertical lines is undefined    
 - The slope of a horizontal line is 0
 - Parallel lines have the ```same slope```   
@ -432,25 +432,25 @@ x, & \text{if  } x > 0\\
 ## Relations
 - A relation can be described using   
   1. Table of Values (see below)   
-   2. Equations (y = 3x + 5)   
+   2. Equations $`(y = 3x + 5)`$   
   3. Graphs (Graphing the equation)    
   4. Words 
 - When digging into the earth, the temperature rises according to the
- following linear equation: t = 15 + 0.01 h. **t** is the increase in temperature in
- degrees and **h** is the depth in meters.
+- following linear equation: $`t = 15 + 0.01 h`$. $`t`$ is the increase in temperature in
+- degrees and $`h`$ is the depth in meters.

 ## Perpendicular Lines
 - To find the perpendicular slope, you will need to find the slope point   
 - Formula: slope1 &times; slope2 = -1   
- Notation: m<sub>&perp;</sub>   
+- Notation: $`m_\perp`$   
 - <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/parallel-perpendicular-lines/parallel_perpendicular_lines_1.gif" width="300">   


 ## Definitions
 - ```Parallel```: 2 lines with the same slope   
 - ```Perpendicular```: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.      
- ```Domain```: The **ordered** set of all possible values of the independent variable (x).   
- ```Range```: The **ordered** set of all possible values of the dependent variable (y).   
+- ```Domain```: The **ordered** set of all possible values of the independent variable $`x`$.   
+- ```Range```: The **ordered** set of all possible values of the dependent variable $`y`$.   
 - ```Continous Data```: A data set that can be broken into smaller parts. This is represented by a ```Solid line```.   
 - ```Discrete Data```: A data set that **cannot** be broken into smaller parts. This is represented by a ```Dashed line```.   
 - ```First Difference```: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.   
@ -525,18 +525,22 @@ x, & \text{if  } x > 0\\
 - <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">   

 ## Discriminant
- The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the 
- coefficients and constant of a quadratic equation: **y = ax<sup>2</sup> + bx + c**    
- D = b<sup>2</sup> - 4ac    
- D > 0 ```(2 distinct real solutions)```     
- D = 0 ```(1 real solution)```    
- D < 0 ```(no real solutions)``` 
+- The discriminant determines the number of solutions (roots) there are in a quadratic equation. $`a, b , c`$ are the 
+- coefficients and constant of a quadratic equation: $`y = ax^2 + bx + c`$    
+ $`
+ D = b^2 - 4ac 
+ \begin{cases}
+ \text{2 distinct real solutions}, & \text{if } D > 0 \\
+ \text{1 real solution}, & \text{if } D = 0 \\
+ \text{no real solutions}, &  \text{if } D < 0 
+ \end{cases}
+ `$

 - <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">

 ## Solving Linear-Quadratic Systems
 - To find the point of intersection, do the following:     
-  1. Isolate both equations for ```y```
+  1. Isolate both equations for $`y`$
  2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
  3. Simplify and put everything on one side and equal to zero on the other side
  4. Factor
@ -547,7 +551,7 @@ x, & \text{if  } x > 0\\
 - <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">      

 - There are 3 possible cases     
- In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
+- In addition, to determine the number of solutions, you the Discriminant formula $`D = b^2 - 4ac`$


 # Ways to solve Systems of Equations
@ -559,14 +563,18 @@ x, & \text{if  } x > 0\\
       y = x + 10  (1)
       x + y + 34 = 40 (2)
       ```
-       We can sub (1) into (2) to find ```x```, then you the value of ```x``` we found to solve for ```y```
-       ```x + (x + 10) + 34 = 40```   
-       ```2x + 44 = 40```    
-       ```2x = -4```   
-       ```x = -2```   
-       Then solve for ```y```   
-       ```y = -2 + 10```   
-       ```y = -8```
+     - We can sub $`(1)`$ into $`(2)`$ to find $`x`$, then you the value of $`x`$ we found to solve for $`y`$
+       ```
+       x + (x + 10) + 34 = 40   
+       2x + 44 = 40    
+       2x = -4   
+       x = -2
+       ```
+     - Then solve for $`y`$   
+       ```
+       y = -2 + 10   
+       y = -8
+       ```

 ## 2. Elimination
   - Here we eliminate a variable by basically eliminate a variable from an equation   
@ -576,7 +584,7 @@ x, & \text{if  } x > 0\\
       2x + 3y = 10 (1)
       4x + 3y = 14 (2)
       ```
-       We can then use elimination
+     - We can then use elimination
       ```
       4x + 3y = 14
       2x + 3y = 10
@ -584,10 +592,12 @@ x, & \text{if  } x > 0\\
       2x + 0 = 4
       x = 2
       ```
-       Then sub the value of ```x``` into an original equation and solve for ```y```   
-       ```2(2) + 3y = 10```   
-       ```3y = 6```   
-       ```y = 2```   
+     - Then sub the value of $`x`$ into an original equation and solve for $`y`$   
+       ```
+       2(2) + 3y = 10   
+       3y = 6   
+       y = 2
+       ```   

 ## 3. Graphing
   - we can rewrite the equations into ```y-intercept form``` and then graph the lines, and see where the lines intersect (P.O.I), and the P.O.I is the solution
@ -598,20 +608,20 @@ x, & \text{if  } x > 0\\

 - |  |Use ```Dash``` line|Use ```Solid line```|   
   |:-|:------------------|:-------------------|   
-   |Shade the region ```above``` the line|y > mx + b|y &ge; mx + b|   
-   |Shade the region ```below``` the line|y < mx + b| y &le; mx + b|    
+   |Shade the region ```above``` the line|$`y > mx + b`$|$`y \ge mx + b`$|   
+   |Shade the region ```below``` the line|$`y < mx + b`$|$`y \le mx + b`$|    

 - ## If   

-  - |x > a <br> x &ge; a|
+  - |$`x > a`$ <br> $`x \ge a`$|
    |:------------------| 
-  - |shade the region on the **right**|  
+    |shade the region on the **right**|  

 - ## If   

-  - |x < a <br> x &le; a|   
-  - |:------------------|   
-  - |shade the region on the **left**|  
+  - |$`x < a`$ <br> $`x \le a`$|   
+    |:------------------|   
+    |shade the region on the **left**|  

 - Step 1. change all inequalities to ```y-intercept form```   
 - Step 2. graph the line