Question 1

Which of the following best describes the solution set for the intersection of two linear inequalities in two variables?

Accepted Answer

The set of all points that satisfy both inequalities at the same time.

Question 2

Given the system of inequalities $4x + y \geq 2$ and $y \leq 1$, which of the following points is in the solution set?

Accepted Answer

$(0, 1)$

Question 3

What is the first algebraic step to solve for $y$ in the inequality $4x + y \geq 2$?

Accepted Answer

Subtract $4x$ from both sides.

Question 4

After rewriting $4x + y \geq 2$ in terms of $y$, what is the equivalent inequality?

Accepted Answer

$y \geq -4x + 2$

Question 5

Which region represents the solution to the system $y \geq -4x + 2$ and $y \leq 1$?

Accepted Answer

The region between the two lines, including the boundaries.

Question 6

If a point $(a, b)$ satisfies $y \geq -4x + 2$ but not $y \leq 1$, what can you conclude?

Accepted Answer

The point is only in the solution set of $y \geq -4x + 2$.

Question 7

Which of the following ordered pairs is NOT in the solution set of $y \geq -4x + 2$ and $y \leq 1$?

Accepted Answer

$(0, 2)$

Question 8

When graphing the intersection of two linear inequalities, what does the darkest shaded region represent?

Accepted Answer

The region where both inequalities are satisfied.

Question 9

Suppose you are given the inequalities $y \leq 1$ and $y \geq -4x + 2$. Which of the following best describes the boundary lines on the graph?

Accepted Answer

Both lines are solid because the inequalities include equality.

Question 10

A real-world application: A company must satisfy the constraints $4x + y \geq 2$ (production requirement) and $y \leq 1$ (resource limit). Which of the following combinations of $x$ and $y$ is feasible?

Accepted Answer

$x = 0, y = 1$

Question 11

If you are asked to verify whether a point is in the solution set of a system of inequalities, what should you do?

Accepted Answer

Check if the point satisfies both inequalities.

Question 12

Which of the following statements is TRUE about the intersection of two linear inequalities?

Accepted Answer

The intersection is the region where both inequalities are satisfied.

Question 13

Given the system $y \geq -4x + 2$ and $y \leq 1$, which of the following is the correct graphical representation?

Accepted Answer

Shade above $y = -4x + 2$ and below $y = 1$; the intersection is between the two lines.

Question 14

If a point lies in the region that satisfies only one of the inequalities in a system, what does this mean?

Accepted Answer

The point is not in the solution set of the system.

Question 15

Which of the following best explains why the intersection of two inequalities is interpreted as 'and'?

Accepted Answer

Because a point must satisfy both inequalities at the same time.

Graphing the Intersection of Two Linear Inequalities 2.4

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