Question

1. when solving real - world problems involving inequalities, what is the significance of the shaded area on a graph?

a. it represents the set of all possible solutions that satisfy the inequality
b. it indicates areas outside the feasible region
c. it has no significant meaning
d. it represents where the inequality is not valid

1. a pet store wants to stock at least 20 animals, choosing between cats and dogs. the number of dogs must be at least twice the number of cats. which system of inequalities describes this situation?

a. ( d + c leq 20; d leq 2c )
b. ( d + c geq 20; d leq 2c )
c. ( d + c geq 20; d geq 2c )
d. ( d - c geq 20; d geq 2c )

1. which graphing method is not appropriate for representing an inequality in two variables?

a. using a dashed line for ( y > )
b. using a double line for ( y < )
c. using a solid line for ( y geq )
d. shading above the line for ( y > )

Explanation:

Question 2

When graphing inequalities, the shaded area represents all the points \((x,y)\) that satisfy the inequality. For example, in \(y>x\), the shaded area above the line \(y = x\) contains all \((x,y)\) pairs where \(y\) is greater than \(x\). Options b, c, and d are incorrect because the shaded area is the feasible (solution - containing) region, it has a clear meaning related to the inequality's solution set, and it represents where the inequality is valid.

Let \(d\) be the number of dogs and \(c\) be the number of cats.

• The store wants to stock at least 20 animals. The phrase "at least" means \(\geq\), so \(d + c\geq20\).
• The number of dogs must be at least twice the number of cats. So \(d\geq2c\)

Options a and b have the wrong inequality for the total number of animals (\(\leq\) instead of \(\geq\)). Option d has the wrong operation (\(d - c\) instead of \(d + c\)) for the total number of animals.

• For \(y>mx + b\), we use a dashed line (because the line \(y=mx + b\) is not included in the solution set), so option a is appropriate.
• For \(y\geq mx + b\), we use a solid line (because the line \(y = mx + b\) is included in the solution set), so option c is appropriate.
• For \(y>mx + b\), we shade above the line (for a line with positive slope; in general, for \(y>mx + b\), if we pick a test - point \((x_0,y_0)\) above the line \(y=mx + b\), \(y_0>mx_0 + b\) holds), so option d is appropriate.
• There is no standard graphing convention of using a double line for \(y < mx + b\). We use a dashed line for \(y

Answer:

a. It represents the set of all possible solutions that satisfy the inequality

Question 3