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Solve the inequality and explain how

Solve the inequality and explain how-example-1
User Wade
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Answer:

B. (see attached)

Explanation:

You want the solution and graph of the inequality |x+3| > 2.

Solution

The inequality resolves to two inequalities with different domain definitions:

|x +3| > 2

When the argument is negative, (x+3) < 0, this is ...

-(x +3) > 2

-x -3 > 2 . . . . . . eliminate parentheses

x +3 < -2 . . . . . . multiply by -1

x < -5 . . . . . . . . . subtract 3; consistent with the domain definition

When the argument is positive, (x+3) > 0, this is ...

x +3 > 2

x > -1 . . . . . . . . subtract 3

-1 < x . . . . . . . . same thing using the < symbol

These differing solution sets do not overlap, but elements of either set are solutions to the inequality. The appropriate conjunction is "OR":

solution: x < -5 or -1 < x

The graph is attached.

__

Additional comment

We like to use the < or ≤ symbols when expressing the solution to an inequality. That way, the relation of the variable to the boundary value is the same as its relation on a number line. Solution values are left of -5 or right of -1:

x < -5 or -1 < x

It helps provide a check that the graph is properly drawn.

The inequality can be rewritten as ...

|x -(-3)| > 2

which can be interpreted as saying the positive distance from x to -3 is more than 2. This tells you the graph will have two disjoint branches, and the appropriate conjunction is OR.

If the problem were different, and the inequality were ...

|x -(-3)| < 2

it would be telling you the solution values have a distance less than 2 from -3. They will be in one continuous band from -5 to -1, so the appropriate conjunction is AND. The solution in this case is usually written as a compound inequality with no conjunction: -5 < x < -1. (Note the use of < symbols puts the variable value in the middle, between the boundary values, as in graph D.)

Solve the inequality and explain how-example-1
User Naqash Malik
by
7.6k points

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