Checking Solutions for Inequalities

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.

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 point sits on the line of 3x+2y<11?

• (2, 3)

• (0, 5)

• (3, 2)

• (1, 4)

Answer explanation

To check which point lies on the line of 3x + 2y < 11, substitute the coordinates into the inequality. For (1, 4): 3(1) + 2(4) = 3 + 8 = 11, which satisfies the equation. Thus, (1, 4) is the correct choice.

When graphing the inequality 3x+y≤8, which region is shaded?

• The region above the line y = -3x + 8 is shaded.

• The region below the line y = -3x + 8 is shaded.

• The region to the right of the line y = -3x + 8 is shaded.

• The region to the left of the line y = 3x - 8 is shaded.

Answer explanation

To graph the inequality 3x + y ≤ 8, first rewrite it as y ≤ -3x + 8. The line y = -3x + 8 is the boundary, and since the inequality is 'less than or equal to', the region below this line is shaded.

What does it mean when a solution point lies on a dashed boundary line?

• The point is not part of the solution set.

• The point is a valid solution point.

• The point indicates a boundary solution.

• The point is part of the solution set.

Answer explanation

A dashed boundary line indicates that points on the line are not included in the solution set. Therefore, if a solution point lies on this line, it is not part of the solution set.

If you have the inequalities -x+2y≥5 and x+3y<4, which point satisfies both?

• (2, 1)

• (0, 3)

• (-4, 2)

• (3, 0)

Answer explanation

To check which point satisfies both inequalities, substitute each point into the inequalities. The point (-4, 2) satisfies -(-4)+2(2)≥5 and -4+3(2)<4, making it the correct choice.

What is a key difference between solving linear inequalities and linear equations in one variable?

• Linear equations can have more than one solution, while inequalities cannot.

• Linear inequalities require checking the solution by substituting back into the original inequality.

• Linear inequalities require flipping the inequality sign when multiplying or dividing by a negative number.

• Linear equations require graphing the solution on a number line.

Which of the following is a TRUE statement about INEQUALITIES?

• Every inequality has only one solution.

• Inequalities may have multiple solutions.

Is this an equation or an inequality?

12x - 7 < 44

• Equation

• Inequality