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Determine which of the ordered pairs listed below satisfies both of the following inequalities:

Y

Y

Y

_{1}< 7Y

_{2}> 3x-1Correct Answer:

**C**Any point with y greater than 7 can be eliminated immediately.

- Construct a truth table, testing the ordered pairs against the individual inequalities. Any pair that is true for both inequalities satisfies both, and is a solution to the given problem. One such table appears below.
- In order to determine the truth value of an ordered pair in the context of an inequality, we substitute the X quantity of the pair in question into the inequality, then compute the output Y value and compare it to the inequality itself. In the case where no such operation is possible, such as with inequality Y
_{1}, we simply compare the Y component of the pair being considered with the inequality. These steps will be specifically detailed immediately following the table.

Y _{1}Y _{2}A(2,9) F T B(3,8) F F C(1,3) T T D(2,4) E(2,3) T F - The ordered pair (2,9) fails to satisfy the first inequality, Y
_{1}< 7, because its Y coordinate, 9, is greater than 7. However, (2,9) does satisfy the second inequality because the substitution of X = 2 into Y_{2}> 3X-1 yields Y_{2}> 5, which is true as 9 is greater than 5. - The ordered pair (3,8) has a Y coordinate greater than 7, thus failing to satisfy Y
_{1}< 7. When the X value of 3 is substituted into Y_{2}, the result is 8, which means that we have also failed to satisfy Y_{2}(remember that Y_{2}specifies a value greater than 3x-1, whereas 8 is equal to 3x-1 when X = 3). - The ordered pair (1,3) satisfies the first inequality, Y
_{1}< 7, as 3 is less than 7. It also satisfies the second inequality, Y_{2}> 3X -1, as substitution of the X value 1 gives the output value 2, and 3 is larger than 2. Since both values are true, both inequalities are satisfied. However, in the spirit of complete work, we will consider the next pair. - The ordered pair (2,4) has a Y coordinate less than 7, and so satisfies Y
_{1}< 7. However, substitution of X = 2 into Y_{2}> 3x-1 yields 5, and since the pair's Y coordinate is less than 5, we fail this test. - The ordered pair (2,3) has a Y coordinate less than 7, and so satisfies Y
_{1}< 7. However, substitution of X = 2 into Y_{2}> 3x-1 yields 5, and since the pair's Y coordinate is less than 5, we fail this test.