Question

How do you solve the inequality `9 > -3 (x - 1) ≥ -12`?

Answer 1

`-2 < x <= 5`

Explanation:

Given: `9 > -3(x-1)>= -12`

`color(blue)("Manipulate the same way you do equation.")`
`color(blue)("but be mindful about the inequality sign")`

Divide throughout by 3 (does not change the signs)

`3 > -x+1 >= -4`

Subtract 1 throughout

`2 > -x >= -5`

`color(brown)("Consider the part: "-x >= -5)`

Swap sides and change the negatives to positives. Keep the inequality the same way round.

Same thing as: add 5 to both sides and add `x` to both sides and keep the inequality the same way round.

Same thing as: multiply both sides by (-1) and turn the inequality sign the other way round.

`5 >= x color(white)("ddd")larr` This is the same thing as `x <= 5`

`color(brown)("Consider the part: "2 > -x)`

Swap sides and change the signs but keep the inequality the same way round.

`x > -2`

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`color(brown)("Putting it all together")`

`-2 < x <= 5`