Math

QuestionSolve the inequality 3z1>z+9|3z - 1| > z + 9.

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of the direction. Therefore, it is always non-negative. . An absolute value inequality is an inequality that contains an absolute value expression.
3. The inequality 3z1>z+9|3z-1| > z+9 can be split into two separate inequalities, depending on whether 3z13z-1 is positive or negative.

STEP 2

We will split the inequality into two cases, one where z1z-1 is positive and one where it is negative.
Case1 z1z-1 is positive or zero, i.e., z10z-1 \geq0. In this case, the inequality becomesz1>z+9z-1 > z+9Case2 z1z-1 is negative, i.e., z1<0z-1 <0. In this case, the inequality becomes(z1)>z+9-(z-1) > z+9

STEP 3

Now, we will solve the inequality from Case1.
Subtract zz from both sides of the inequality3zz>9+13z - z >9 +1

STEP 4

implify the inequality2z>102z >10

STEP 5

Divide both sides of the inequality by2z>5z >5

STEP 6

Now, we will solve the inequality from Case2.
First, distribute the negative sign on the left side of the inequality3z+1>z+9-3z +1 > z +9

STEP 7

Subtract zz and1 from both sides of the inequality3zz>91-3z - z >9 -1

STEP 8

implify the inequality4z>8-4z >8

STEP 9

Divide both sides of the inequality by -4. Remember, when we divide or multiply an inequality by a negative number, the direction of the inequality changesz<2z < -2

STEP 10

Now, we have two solutions from the two cases. However, we need to check if these solutions satisfy the original inequality.The solutions are z>5z >5 and z<2z < -2.We can see that these two solutions do not overlap, so the solution to the original inequality 3z>z+9|3z-| > z+9 isz<2orz>5z < -2 \, \text{or} \, z >5

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