The inequalities that require flipping the sign are:
-5x - 10 > 5
[tex]\frac{x}{-7} + 3 \leq 4[/tex]
Solution:
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
option 1)
-5x - 10 > 5
Move -10 from L.H.S to R.H.S
-5x > 5 + 10
-5x > 15
Divide the above expression by 5
[tex]-x > 3[/tex]
Divide the above inequality by -1, so we must flip the sign
x < -3
option 2)
[tex]7x - 5 \leq 16[/tex]
Move the constant term from L.H.S to R.H.S
[tex]7x \leq 16 + 5\\\\7x \leq 21\\\\[/tex]
Divide the above inequality by 7
[tex]x \leq 3[/tex]
This does not required flipping the symbol
option 3
[tex]\frac{x}{5} - 6 > -11[/tex]
Move the constant term from L.H.S to R.H.S
[tex]\frac{x}{5} > -11 + 6\\\\\frac{x}{5} > -5[/tex]
Multiply both the sides by 5
[tex]x > -25[/tex]
This does not required flipping the symbol
option 4
[tex]x + 12 \leq 29[/tex]
Move the constant term from L.H.S to R.H.S
[tex]x \leq 29 - 12\\\\x \leq 17[/tex]
This does not required flipping the symbol
option 5
[tex]\frac{x}{-7} + 3 \leq 4[/tex]
Move the constant term from L.H.S to R.H.S
[tex]\frac{x}{-7} \leq 4-3\\\\\frac{x}{-7} \leq 1\\\\[/tex]
Multiply both the sides by -7, so we must flip the sign
[tex]x \geq -7[/tex]
Thus this requires flipping the sign
