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QuestionSolve the inequality −3∣x+2∣<−9-3|x+2|<-9.

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is always non-negative, i.e., it is either positive or zero. . The inequality −3∣x+∣<−9-3|x+|<-9 is a compound inequality, which means it can be split into two separate inequalities.

STEP 2

First, we need to isolate the absolute value expression. We can do this by dividing both sides of the inequality by -. Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
−∣x+2∣−>−9−\frac{-|x+2|}{-}>\frac{-9}{-}

STEP 3

implify the inequality.
∣x+2∣>3|x+2|>3

STEP 4

The absolute value inequality ∣x+2∣>3|x+2|>3 can be rewritten as a compound inequality. The compound inequality is x+2>3x+2>3 or x+2<−3x+2<-3.

STEP 5

olve the first inequality x+2>3x+2>3 by subtracting2 from both sides.
x+2−2>3−2x+2-2>3-2

STEP 6

implify the first inequality.
x>1x>1

STEP 7

olve the second inequality x+2<−3x+2<-3 by subtracting2 from both sides.
x+2−2<−3−2x+2-2<-3-2

STEP 8

implify the second inequality.
x<−5x<-5The solution to the inequality −3∣x+2∣<−-3|x+2|<- is x>1x>1 or x<−5x<-5.

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