<em><u>The inequalities that require flipping the sign are:</u></em>
-5x - 10 > 5
![\frac{x}{-7} + 3 \leq 4](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B-7%7D%20%2B%203%20%5Cleq%204)
<em><u>Solution:</u></em>
Let us the inequalities one by one
You can perform on operations on both sides of inequality, and have its truth value unchanged
But if we multiply or divide by a negative number , we must flip the sign
<h3>option 1) </h3>
<em><u>-5x - 10 > 5</u></em>
Move -10 from L.H.S to R.H.S
-5x > 5 + 10
-5x > 15
Divide the above expression by 5
![-x > 3](https://tex.z-dn.net/?f=-x%20%3E%203)
Divide the above inequality by -1, so we must flip the sign
x < -3
<h3>option 2)</h3>
![7x - 5 \leq 16](https://tex.z-dn.net/?f=7x%20-%205%20%5Cleq%2016)
Move the constant term from L.H.S to R.H.S
![7x \leq 16 + 5\\\\7x \leq 21\\\\](https://tex.z-dn.net/?f=7x%20%5Cleq%2016%20%2B%205%5C%5C%5C%5C7x%20%5Cleq%2021%5C%5C%5C%5C)
Divide the above inequality by 7
![x \leq 3](https://tex.z-dn.net/?f=x%20%5Cleq%203)
This does not required flipping the symbol
<h3>option 3</h3>
![\frac{x}{5} - 6 > -11](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B5%7D%20-%206%20%3E%20-11)
Move the constant term from L.H.S to R.H.S
![\frac{x}{5} > -11 + 6\\\\\frac{x}{5} > -5](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B5%7D%20%3E%20-11%20%2B%206%5C%5C%5C%5C%5Cfrac%7Bx%7D%7B5%7D%20%3E%20-5)
Multiply both the sides by 5
![x > -25](https://tex.z-dn.net/?f=x%20%3E%20-25)
This does not required flipping the symbol
<h3>option 4</h3>
![x + 12 \leq 29](https://tex.z-dn.net/?f=x%20%2B%2012%20%5Cleq%2029)
Move the constant term from L.H.S to R.H.S
![x \leq 29 - 12\\\\x \leq 17](https://tex.z-dn.net/?f=x%20%5Cleq%2029%20-%2012%5C%5C%5C%5Cx%20%5Cleq%2017)
This does not required flipping the symbol
<h3>
option 5</h3>
![\frac{x}{-7} + 3 \leq 4](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B-7%7D%20%2B%203%20%5Cleq%204)
Move the constant term from L.H.S to R.H.S
![\frac{x}{-7} \leq 4-3\\\\\frac{x}{-7} \leq 1\\\\](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B-7%7D%20%5Cleq%204-3%5C%5C%5C%5C%5Cfrac%7Bx%7D%7B-7%7D%20%5Cleq%201%5C%5C%5C%5C)
Multiply both the sides by -7, so we must flip the sign
![x \geq -7](https://tex.z-dn.net/?f=x%20%5Cgeq%20-7)
Thus this requires flipping the sign