Solve each inequality. Graph the solution. |3 z|-4>8 |...

Name: Problem 21, Chapter 1: Algebra 2
Uploaded: 2020-03-20T00:41:15-08:00
Duration: 3 min 4 s
Channel: Kelsey Van Horn
Description: Solve each inequality. Graph the solution. |3 z|-4>8

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve an expression inside an absolute value set in relation to a number, and they require considering two cases to cover the full range of possible values. For inequalities of the form |expression| > constant, the solution is derived from both expression > constant and expression < -constant, provided that constant is positive.

Absolute Value Function

The absolute value function measures the distance of a number from zero, regardless of its sign. It is defined as |x|, which equals x if x is positive or zero, and -x if x is negative. Understanding this concept is fundamental when dealing with any expression involving absolute values.

Graphing Inequality Solutions

Graphing the solutions of inequalities typically involves plotting the critical points on a number line and then shading the appropriate regions that satisfy the inequality. This visual representation helps to confirm the range of values that meet the inequality condition.

Transcript

00:01 Okay, so this problem asks us to solve the inequality of the absolute value of 3z minus 4 is greater than 8, and then we are asked to graph the solution.

00:10 So to start, we want to simplify this inequality by getting the absolute value portion on one side of the equation by itself.

00:17 So we'll start by adding 4 to each side.

00:19 And that gives us the inequality of the absolute value of 3z is greater than 12.

00:25 And from here, we'll split it into two different inequalities to account for the fact that what's inside the this absolute value portion can be either positive or negative because the absolute value of a negative number is still positive.

00:38 So the two inequalities will get will be 3z is greater than 12.

00:43 And then the second one, we're just going to multiply what's inside this absolute value symbol by negative 1.

00:50 And so that gives us a second inequality of negative 3z is greater than 12.

00:57 And now we just have to solve each inequality for z.

01:00 So over here on the 1 ,000, left, we'll divide each side by three, and we get z is greater than four.

01:07 And over here on the right, we'll divide each side by negative three.

01:10 Now, when you divide or multiply an inequality by a negative number, you have to flip the sign.

01:16 So this will give us the inequality of z is less than negative four.

01:20 Then we're asked to graph the solution.

01:23 So when we graph the solution, we're going to put it on a number line.

01:28 And our two potential answers, we'll start with, we'll do two number lines, so we can get a better representation, but we'll imagine this, imagine them as being one long continuous number line.

01:40 So the first one we will graph will be that z is less than negative 4.

01:47 So if here's 0, and then we'll have negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, and so on, all the way for all negative values...

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