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# Inequalities: Simultaneous Inequalities

If $$-13 < 7a + 1 < 29$$ and $$19 < 2 - b < 23$$, what is the maximum possible integer value of $$a+b$$?

Correct.

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In this case, $$7a + 1$$ and $$-b$$ get in the way of reaching the desired result of $$a + b$$. Subtract $$1$$ from the first inequality to isolate $$a$$. >$$-13 - 1 < 7a + 1 - 1 < 29 - 1$$ >$$-14 < 7a < 28$$ Now, divide by $$7$$ to isolate $$a$$. >$$\frac{-14}{7} < \frac{7a}{7} < \frac{28}{7}$$ >$$-2 < a < 4$$ Subtract $$2$$ from the second inequality. >$$19 - 2 < 2 - b - 2 < 23 - 2$$ >$$17 < -b < 21$$ Multiply the second inequality by $$-1$$ to isolate $$b$$. Don't forget to flip the sign. >$$17 \cdot (-1) > -b \cdot (-1) > 21 \cdot (-1)$$ >$$-17 > b > -21$$ Now, line the inequalities up so that the sign goes in the same direction. >$$-2 < a < 4$$ >$$-21 < b < -17$$ Finally, add the inequalities. >$$-23 < a + b < -13$$ Since we're looking for the integer value of the sum, which must be smaller than $$-13$$, go down one integer to $$-14$$.

Incorrect.

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If you've done your calculations correctly, you must have reached the conclusion that $$a+b$$ must be smaller than $$-13$$. Don't be hasty, though. Is $$-12$$ really smaller than $$-13$$?

Incorrect.

If you've done your calculations correctly, you must have reached the conclusion that $$a+b$$ must be smaller than $$-13$$, which means that the sum cannot equal $$-13$$. Look for a smaller answer. [[snippet]]

Incorrect.

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Incorrect.

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$$-23$$
$$-18$$
$$-14$$
$$-13$$
$$-12$$