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For each inequality, find the critical numbers and fill them in at the proper spots on the number line (note that the scale on the number line does not matter, just the relative position of the critical numbers. Then write in which bracket belongs at each critical position, and check off which regions should be shaded.

Solve each inequality for

. Write solutions in interval notation.a)

Critical values: and

Test intervals: , ,

For , I'll try the value of for .

is not less than or equal to , so the interval is not part of the solution set for .

For , I'll try the value of for .

is less than , satisfying our original inequality, so the interval is in the solution set for .

For , I'll try the value of for .

is not less than or equal to , so the interval is not part of the solution set for .

Now let's test the critical values.

is equal to , so is part of the solution set.

is equal to , so is part of the solution set.

That means that the solution is the interval .

b)

The critical values of are and .

Test intervals: , ,

For , I'll try the value of for .

is greater than , so the interval is part of the solution set for .

For , I'll try the value of for .

is not greater than , so is not part of the solution set for .

For , I'll try the value of for .

is greater than , so is part of the solution set for .

Now let's test the critical values.

is not greater than , so is not part of the solution set.

Since the denominator of the fraction is , and we can't divide by , is not part of the solution set.

That means that the solution is .

Solve each inequality for

a)

Critical values: and

Test intervals: , ,

Testing for :

is greater than , so is part of the solution set.

Testing for :

is not greater than or equal to , so is not part of the solution set.

Testing for :

is greater than , so is part of the solution set.

Now let's test the critical values, and .

is equal to , so is part of the solution set.

is equal to , so is part of the solution set.

That means that the solution is .

b)

Critical values: and

Test intervals: , ,

Testing for :

is not less than or equal to , so is not part of the solution set.

Testing for :

is less than , so is part of the solution set.

Testing for :

c)

Critical values:

Test intervals: ,

Testing for :

is not greater than , so is not part of the solution set.

Testing for :

is greater than , so is part of the solution set.

Testing the critical value, :

is not greater than , so is not part of the solution set.

That means that the solution is .

d)

Critical values: and

Test intervals: , ,

Testing for :

is less than , so is part of the solution set.

Testing for :

is not less than , so is not part of the solution set.

Testing for :

is less than , so is part of the solution set.

Testing the critical values, and :

is not less than , so neither nor is part of the solution set.

That means that the solution is .

e)

Critical values: and

Test intervals: , ,

Testing for :

is not less than , so is not part of the solution set.

Testing for :

is less than , so is part of the solution set.

Testing for :

is not less than , so is not part of the solution set.

Testing the critical values, and :

is not less than , so neither nor is part of the solution set.

That means that the solution is .

f)

Critical values: and

Test intervals: , ,

Testing for :

is greater than , so is part of the solution set.

Testing for :

is less than , so is not part of the solution set.

Testing for :

is greater than , so is part of the solution set.

That means that the solution is .