SAT Question 830120b0: Answer, Explanation & Desmos Solution

$y > 2 x - 1$

$2 x > 5$

Which of the following consists of the $y$ -coordinates of all the points that satisfy the system of inequalities above?

Difficulty:Hard

$y > 6$

$y > 4$

$y > \frac{5}{2}$

$y > \frac{3}{2}$

Explanation

Correct answer: B.

Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting $1$ from each side of the inequality $2 x > 5$ gives $2 x - 1 > 4$ . So the given system of inequalities is equivalent to the system of inequalities $y > 2 x - 1$ and $2 x - 1 > 4$ , which can be rewritten as $y > 2 x - 1 > 4$ . Using the transitive property of inequalities, it follows that $y > 4$ .

Desmos solution:

Note: Use x=5/2 as the excluded boundary in y>2x-1, so the y-coordinates must be greater than 4.

Enter in Desmos:

$r = 2 (\frac{5}{2}) - 1$
$y > 4$

Reviewed and rewritten by CookSAT

Why the other choices are wrong

Choice A

Choice A is incorrect because there are points with a $y$ -coordinate less than $6$ that satisfy the given system of inequalities. For example, the point $(3, 5.5)$ satisfies both inequalities.

Choice C

Choice C is incorrect. This may result from solving the inequality $2 x > 5$ for $x$ , then replacing $x$ with $y$ .

Choice D

Choice D is incorrect because this inequality allows $y$ -values that are not the $y$ -coordinate of any point that satisfies both inequalities. For example, $y = 2$ is contained in the set $y > \frac{3}{2}$ ; however, if $2$ is substituted into the first inequality for $y$ , the result is $x < \frac{3}{2}$ .

SAT Question 830120b0: Linear Inequalities answer and explanation with Desmos solution

Explanation

Enter in Desmos:

Why the other choices are wrong