Math

QuestionSolve for xx: 12<14(x+3)<1512 < -14(x+3) < 15. Type DNE if no solution exists. Provide your answer in interval notation.

Studdy Solution

STEP 1

Assumptions1. We are solving for xx in the inequality 12<14(x+3)<1512<-14(x+3)<15. . The inequality 12<14(x+3)<1512<-14(x+3)<15 is a compound inequality, which means it is two inequalities combined into one statement.
3. We will solve each part of the compound inequality separately and then combine the solutions.

STEP 2

The compound inequality can be broken down into two separate inequalities12<14(x+)12<-14(x+)and14(x+)<15-14(x+)<15

STEP 3

Let's solve the first inequality 12<14(x+3)12<-14(x+3).
First, distribute the 14-14 to both xx and 33.
12<14x4212<-14x-42

STEP 4

Next, add 4242 to both sides of the inequality to isolate the 14x-14x term.
12+42<14x12+42<-14x

STEP 5

implify the left side of the inequality.
54<14x54<-14x

STEP 6

Divide both sides of the inequality by 14-14 to solve for xx. Remember that when you divide or multiply an inequality by a negative number, you must flip the inequality sign.
x>5414x>\frac{54}{-14}

STEP 7

implify the right side of the inequality.
x>3.857142857142857x>-3.857142857142857

STEP 8

Now, let's solve the second inequality 14(x+3)<15-14(x+3)<15.
First, distribute the 14-14 to both xx and 33.
14x42<15-14x-42<15

STEP 9

Next, add 4242 to both sides of the inequality to isolate the 14x-14x term.
14x<15+42-14x<15+42

STEP 10

implify the right side of the inequality.
14x<57-14x<57

STEP 11

Divide both sides of the inequality by 14-14 to solve for xx. Remember that when you divide or multiply an inequality by a negative number, you must flip the inequality sign.
x>5714x>\frac{57}{-14}

STEP 12

implify the right side of the inequality.
x<4.071428571428571x<-4.071428571428571

STEP 13

Now, we combine the solutions of the two inequalities. The solution to the compound inequality is the intersection of the solutions to the two inequalities.
The solution to the compound inequality is x>3.857142857142857x>-3.857142857142857 and x<.071428571428571x<-.071428571428571.

STEP 14

However, there is no value of xx that can satisfy both of these inequalities at the same time. Therefore, the solution to the compound inequality does not exist.
The solution to the compound inequality is DNE.

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