Checking solutions of two-variable inequalities (Math Masters)

What is the general form of a two-variable inequality?

• Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C

• Ax + By + C > 0

• Ax + By ≠ C

• Ax + By = C

Answer explanation

The general form of a two-variable inequality includes expressions like Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C. These represent different relationships between the variables and a constant.

Which of the following points is a solution to the inequality y>3x+4?

• (0, 5)

• (3, 4)

• (1, 2)

• (2, 3)

Answer explanation

To solve the inequality y > 3x + 4, substitute the points. For (0, 5): 5 > 3(0) + 4, which is true. For (3, 4): 4 > 3(3) + 4 is false. For (1, 2) and (2, 3), both are false. Thus, (0, 5) is the only solution.

How do you determine if a point is a solution to a two-variable inequality?

• Graph the inequality and find the intersection point.

• Substitute the point into the inequality and check if the statement is true.

• Choose any random point and assume it is a solution.

• Check if the point lies on the boundary line only.

Answer explanation

To determine if a point is a solution to a two-variable inequality, substitute the point into the inequality. If the resulting statement is true, then the point is a solution. This method directly verifies the point's validity.

What does the boundary line represent in a two-variable inequality?

• The boundary line shows the area where solutions are not valid.

• The boundary line represents the points where the corresponding equation is equal.

• The boundary line indicates the maximum value of the inequality.

• The boundary line represents the average of the two variables.

Answer explanation

The boundary line in a two-variable inequality represents the points where the corresponding equation is equal. It separates the solution region from the non-solution region, indicating where the inequality holds true.

For the inequality y≤x-4 which of the following points is NOT a solution?

• (2, -3)

• (5, 1)

• (3, 0)

• (0, -5)

Answer explanation

To check which point is NOT a solution for y≤x-4, substitute each point. For (3, 0): 0 ≤ 3-4 is false. Thus, (3, 0) is NOT a solution, while the others satisfy the inequality.

If the inequality is y<1x/2+1, what is the correct way to graph it?

• Graph the line y = (1/2)x + 1 as a dashed line and shade below it.

• Graph the line y = (1/2)x + 1 as a dashed line and shade above it.

• Graph the line y = (1/2)x + 1 and do not shade any area.

• Graph the line y = (1/2)x + 1 as a solid line and shade above it.

Answer explanation

To graph the inequality y < (1/2)x + 1, first graph the line y = (1/2)x + 1 as a dashed line, indicating that points on the line are not included. Then, shade below the line to represent all y-values less than the line.

Which of the following statements about the solution set of a two-variable inequality is true?

• The solution set is a single point in the coordinate plane.

• The solution set can only be represented as a line segment.

• The solution set is a region in the coordinate plane.

• The solution set is always empty for two-variable inequalities.

Answer explanation

The correct choice is that the solution set is a region in the coordinate plane. Two-variable inequalities typically represent areas, such as half-planes, rather than just points or line segments.

For the inequality 5x-2y>15, which point is a solution?

• (4, -1)

• (1, 4)

• (5, 10)

• (2, 3)

Answer explanation

To check which point satisfies the inequality 5x - 2y > 15, substitute each point. For (4, -1): 5(4) - 2(-1) = 20 + 2 = 22 > 15. Thus, (4, -1) is a solution. The other points do not satisfy the inequality.

What type of line is used for the inequality y<2x-1?

• Dashed line

• Solid line

• Curved line

• Dotted line

Answer explanation

For the inequality y < 2x - 1, a dashed line is used to indicate that points on the line are not included in the solution set. This is because the inequality is strict (less than), meaning y cannot equal 2x - 1.

Which of the following inequalities represents the shaded region above the line y=-3x+5?

• y > -3x + 5

• y < -3x + 5

• y = -3x + 5

• y ≥ -3x + 5

Answer explanation

The inequality y > -3x + 5 represents the region above the line y = -3x + 5. The other options either include the line or represent the area below it, making them incorrect.