Math

QuestionSolve and graph the compound inequality: 2<3x66<4-2<\frac{3 x-6}{6}<4.

Studdy Solution

STEP 1

Assumptions1. The inequality is a compound inequality, which means it consists of two inequalities combined into one statement by the word "and" or the word "or". . In this case, the compound inequality is "- < (3x -6)/6 <4", which means we are looking for all x such that - is less than (3x -6)/6 and (3x -6)/6 is less than4.

STEP 2

We will solve the compound inequality by first isolating the x term. To do this, we will multiply all parts of the inequality by6 to eliminate the denominator.
2×6<(x6)<4×6-2 \times6 < (x -6) <4 \times6

STEP 3

Calculate the multiplication to simplify the inequality.
12<3x6<24-12 <3x -6 <24

STEP 4

Next, we will add6 to all parts of the inequality to isolate the3x term.
12+6<3x<24+6-12 +6 <3x <24 +6

STEP 5

Calculate the addition to simplify the inequality.
<3x<30- <3x <30

STEP 6

Finally, we will divide all parts of the inequality by3 to solve for x.
63<x<303\frac{-6}{3} < x < \frac{30}{3}

STEP 7

Calculate the division to find the solution for x.
2<x<10-2 < x <10

STEP 8

Now, we will graph the solution on a number line. Draw a number line and mark the points -2 and10. Since the inequality is strict (i.e., it does not include the endpoints), we use open circles at -2 and10. Then, shade the region between -2 and10 to represent all the numbers that are greater than -2 and less than10.

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