Question

Which system of linear inequalities has the point (3, –2) in its solution set?

y less-than negative 3. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.
y greater-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y less-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y greater-than negative 2. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal at y = negative 2. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.

Answer 1

Answer:

The second option will be the answer.

Step-by-step explanation:

We have to select the system of linear inequalities from options which has the point (3,-2) in its solution set.

So, y = - 2 will be within the solution set of the system.

Now, only the second option which gives one of the inequalities of the system i.e. y > - 3 is the right option.

It also gives the second inequality of the system as $y \geq \frac{2}{3}x - 4$.

Now, both the inequalities are satisfied by the point (3,-2).

Therefore, the second option will be the answer. (Answer)

Answer 2

The system of linear inequalities $y > -3$ and $y \ge \frac 23x -4$ has (3,-2) in its solution set.

The solution set is given as:

$(x,y) = (3,-2)$

The above solution set is true for $y > -3$ and $y \ge \frac 23x -4$, and the proof is as follows:

Substitute 3 for x and -2 for y in $y > -3$ and $y \ge \frac 23x -4$

So, we have:

$\mathbf{y > -3}$

$\mathbf{-2 > -3}$ --- this is true, because -2 is greater than -2

Also, we have:

$\mathbf{y \ge \frac 23x -4}$

$\mathbf{-2 \ge \frac 23 \times 3 -4}$

$\mathbf{-2 \ge 2 -4}$

$\mathbf{-2 \ge -2}$ --- this is also true, because -2 is greater than or equal to -2

Hence, the system of linear inequalities $y > -3$ and $y \ge \frac 23x -4$ has (3,-2) in its solution set.

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