Math

QuestionSolve the inequality 3z1>z+9|3z - 1| > z + 9 and graph the solution set.

Studdy Solution

STEP 1

Assumptions1. The given inequality is 3z1>z+9|3z-1| > z+9 . We need to solve this inequality and graph the solution set

STEP 2

An absolute value inequality can be rewritten as two separate inequalities. We can rewrite the given inequality as followsz1>z+9z-1 > z+9andz1<(z+9)z-1 < -(z+9)

STEP 3

We will solve each inequality separately. Let's start with the first inequality3z1>z+93z-1 > z+9

STEP 4

To isolate zz on one side, we subtract zz from both sides3zz>1+93z - z >1 +9

STEP 5

implify the inequality2z>102z >10

STEP 6

Divide both sides by2 to solve for zzz>5z >5

STEP 7

Now we will solve the second inequality3z1<(z+9)3z-1 < -(z+9)

STEP 8

istribute the negative sign on the right side3z1<z3z-1 < -z -

STEP 9

Add zz to both sides to isolate zz3z+z<93z + z < - -9

STEP 10

implify the inequality4z<104z < -10

STEP 11

Divide both sides by4 to solve for zzz<104z < -\frac{10}{4}

STEP 12

implify the inequalityz<52z < -\frac{5}{2}

STEP 13

Now we have two inequalitiesz>5z >5andz<52z < -\frac{5}{2}

STEP 14

To graph the solution set, we draw a number line and mark the points z=z = and z=2z = -\frac{}{2}.

STEP 15

Since z>5z >5, we shade the region to the right of 55.

STEP 16

Since z<52z < -\frac{5}{2}, we shade the region to the left of 52-\frac{5}{2}.
The solution set is the union of these two regions.

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