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# Inequalities: Inequalities Involving an Absolute Value - The Number Case

If $$|3s + 7| > 5$$, which of the following gives all possible values of $$s$$?

Incorrect.

[[Snippet]]

This answer choice is only partially true. It does not define all possible values of $$s$$. Hence, this is not the correct answer.

Incorrect.

[[Snippet]]

This answer choice is only partially true. It does not define all possible values of $$s$$. Hence, this is not the correct answer.

Incorrect.

[[Snippet]]

A possible value of $$s$$ with this answer choice would be $$s = -3$$. Plug this into the original expression, and we have >$$|3s + 7|$$ >$$= |3 \cdot (-3) + 7|$$ >$$= |{-9} + 7|$$ >$$= |{-2}|$$ >$$= 2$$. But 2 is not greater than 5, as required in the question stem, so this cannot be the correct answer.
Correct. Solve absolute values of the number case by considering two possible scenarios.

First scenario: Copy the inequality without the absolute value brackets and solve. >$$3s + 7 > 5$$ >$$3s > 5 - 7$$ >$$3s > -2$$ Divide both sides by 3 to isolate $$s$$. >$$\frac{3s}{3} > -\frac{2}{3}$$ >$$s > -\frac{2}{3}$$

Second scenario: Remove the absolute value brackets. Put a negative sign around the other side of the inequality AND flip the sign. >$$3s + 7 <-5$$ >$$3s < -5 - 7$$ >$$3s < -12$$ Divide both sides by 3 to isolate $$s$$. >$$\frac{3s}{3} < -\frac{12}{3}$$ >$$s < -4$$ Finally, combine the two scenarios into one range for $$s$$: $$-\frac{2}{3} < s$$ or $$s <-4$$. Hence, this is the correct answer.

Incorrect. [[snippet]] Carefully check your calculations.
$$\frac{2}{3} < s< 4$$
$$s<-4$$ or $$s > -\frac{2}{3}$$
$$-4< s<-\frac{2}{3}$$
$$s>-\frac{2}{3}$$
$$s>-4$$