When solving inequalities, you can treat it like you would an equation with an equal sign. However, remember that whenever you multiply or divide by a negative number, you must flip the inequality sign.
You're given [tex] \frac{-ax}{b} \geq c -d [/tex] and asked to solve for x:
1) Multiply both sides by b
[tex] \frac{-ax}{b} \geq c -d \\
-ax \geq b(c-d)[/tex]
2) Distribute the b to the c and d using the distributive property
[tex]-ax \geq b(c-d)\\
-ax \geq b(c)-b(d)\\
-ax \geq bc-bd[/tex]
3) Divide both sides by -a. Don't forget to flip the inequality sign!
[tex]-ax \geq bc-bd\\
x \leq \frac{bc-bd}{-a} [/tex]
4) Simplify by multiplying the right side by [tex] \frac{-1}{-1} [/tex] (aka 1) to get rid of the negative on the a. You don't need to flip the sign here (think of it as dividing and the multiplying by -1, so you flip it twice, which is the same as not flipping at all!)
[tex]x \leq \frac{bc-bd}{-a} \\
x \leq \frac{bc-bd}{-a} ( \frac{-1}{-1} )\\
x \leq \frac{-bc+bd}{a} \\
x \leq \frac{bd-bc}{a}[/tex]
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Answer: Top right, [tex]x \leq \frac{bd-bc}{a}[/tex]
