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

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The inequality $|3z-1|>z+9$ is a complex inequality, where $z$ is a complex number.
3. We are asked to graph the solution set, which means we need to find all values of $z$ that satisfy the inequality and represent them on a complex plane.

STEP 2

The absolute value of a complex number $z=a+bi$ is defined as $\sqrt{a^2+b^2}$ . Therefore, we can rewrite the inequality as follows $\sqrt{(z-1)^2} > z+9$

STEP 3

Square both sides of the inequality to remove the square root. This gives us $(3z-1)^2 > (z+9)^2$

STEP 4

Expand both sides of the inequality $9z^2 -6z +1 > z^2 +18z +81$

STEP 5

Rearrange the inequality to bring all terms to one side $8z^2 -24z -80 >0$

STEP 6

This is a quadratic inequality. To solve it, we first find the roots of the corresponding quadratic equation $8z^2 -24z -80 =0$ .

STEP 7

The roots of a quadratic equation $ax^2 + bx + c =0$ are given by the formula $z = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .

STEP 8

Substitute $a =8$ , $b = -24$ , and $c = -80$ into the formula to find the roots $z = \frac{-(-24) \pm \sqrt{(-24)^2 -48(-80)}}{2*8}$

STEP 9

Calculate the roots $z = \frac{24 \pm \sqrt{576 +256}}{16}$ $z = \frac{24 \pm \sqrt{3136}}{16}$ $z = \frac{24 \pm56}{16}$ So, the roots are $z =5$ and $z = -2$ .

STEP 10

The solution to the quadratic inequality $8z^2 -24z -80 >0$ is the set of all $z$ that are not between the roots. Therefore, the solution set is $z < -2$ or $z >5$ .

STEP 11

To graph the solution set, we draw a number line and mark the roots $-$ and $5$ . We then shade the regions that represent the solution set, which are $z < -$ and $z >5$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms