Finding a Pair of Values that Satisfies Multiple Inequalities

2.14.2 Finding a Pair of Values that Satisfies Multiple Inequalities

2.14.2 • Finding a Pair of Values that Satisfies Multiple Inequalities

Activity

To make a quilt, a quilter is buying fabric in two colors, light and dark. He needs at least 9.5 yards of fabric in total.

The light color costs $9 a yard. The dark color costs $13 a yard. The quilter can spend up to $110 on fabric.

Intricate quilt pattern featuring a striking array of red, maroon, and cream geometric shapes, creating a dynamic and visually rich textile with delicate quilting stitches.

Here are two graphs that represent the two constraints.

Length	Cost

Graphs illustrating inequalities for fabric length and cost needed for a quilt, with respective shaded regions indicating solutions.

Write an inequality to represent the length constraint. Let $x$ represent the yards of light fabric and $y$ represent the yards of dark fabric.

Compare your answer: You answer may vary, but here is a sample.

$x + y \geq 9.5$

Select three pairs that satisfy the length constraint.

$(5, 5)$
$(2.5, 4.5)$
$(7.5, 3.5)$
$(12, 10)$

$(5, 5)$ , $(7.5, 3.5)$ , and $(12, 10)$ .

Write an inequality to represent the cost constraint.

Compare your answer: Your answer may vary, but here is a sample.

$9 x + 13 y \leq 110$

Select three the pairs that satisfy the cost constraint.

$(4, 5)$
$(10, 1)$
$(8, 3)$
$(1, 1)$

$(1, 1)$ , $(4, 5)$ , and $(10, 1)$ .

Explain why $(2, 2)$ satisfies the cost constraint but not the length constraint.

Compare your answer: Your answer may vary, but here is a sample.

2 yards of each color costs $44, which is less than $110, so it satisfies this constraint. However, this is only 4 yards altogether, which is not at least 9.5 yards.

Find at least one pair of numbers that satisfies both constraints. Be prepared to explain how you know.

Compare your answer: Your answer may vary, but here is a sample.

$(10, 1)$ , $(8, 2)$ , and ( $6.5, 3.5)$ (any point that is in the intersection of the two inequalities).

What does the pair of numbers represent in this situation?

Compare your answer: Yoru answer may vary, but here is a sample.

$(10, 1)$ means 10 yards of light fabric and 1 yard of dark fabric. This is 11 yards of fabric altogether, which is more than 9.5. It costs $103 because $9 (10) + 1 (13) = 103$ . This is less than the $110 maximum.

Write the two inequalities together as a system of inequalities.

Compare your answer: Your answer may vary, but here is a sample.

${\begin{matrix} 9 x + 13 y \leq 110 \\ x + y \geq 9.5 \end{matrix}$

Self Check

A singer sold two types of tickets to a performance, seated tickets and standing-room tickets.

Seated tickets cost $25.
Standing-room tickets cost $15.
More than 60 tickets were sold.
More than $1,200 was made from selling tickets.

Which of the following ticket combinations is possible given the situation described here?

20 seated tickets and 50 standing-room tickets
10 seated tickets and 60 standing-room tickets
40 seated tickets and 19 standing-room tickets
14 seated tickets and 55 standing-room tickets

Additional Resources

Determine Whether an Ordered Pair Is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

System of Linear Inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

${\begin{matrix} x + 4 y \geq 10 \\ 3 x - 2 y < 12 \end{matrix}$

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs $(x, y)$ that make both inequalities true.

Solutions of A System of Linear Inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the $(x, y)$ coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Example

Determine whether the ordered pair is a solution to this system:

${\begin{matrix} x + 4 y \geq 10 \\ 3 x - 2 y < 12 \end{matrix}$

a.) $(- 2, 4)$ b.) $(3, 1)$

a.) Is the ordered pair $(- 2, 4)$ a solution?

Text shows substituting x equals negative 2 and y equals 4 into the inequalities x plus 4 times y greater than or equal to 10 and into the inequality 3 times x minus 2 times y less than 12. Both inequalities are verified: 14 greater than or equal to 10 is true and negative 14 less than 12 is true.

The ordered pair $(- 2, 4)$ made both inequalities true. Therefore $(- 2, 4)$ is a solution to this system.

b.) Is the ordered pair $(3, 1)$ a solution?

Text explains substituting x equals 3 and y equals 1 into two inequalities: x plus 4 times y is less than or equal to 10 and the inequality 3 times x minus 2 times y less than 12. The results are 7 greater than or equal to 10 is false and 7 less than 12 is true.

The ordered pair $(3, 1)$ made one inequality true, but it made the other one false. Therefore, $(3, 1)$ is not a solution to this system.

Try it

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

Determine whether the ordered pair is a solution to the system:

${\begin{matrix} y > 4 x - 2 \\ 4 x - y < 20 \end{matrix}$

a. $(- 2, 1)$

b. $(4, - 1)$

a. Yes.

Here is how to determine if each ordered pair is a solution:

Step 1 - Choose an ordered pair.
$(- 2, 1)$

Step 2 - Substitute into the inequality.
$y > 4 x - 2$
$1 > 4 (- 2) - 2$

$4 x - y < 20$
$4 (- 2) - 1 < 20$

Step 3 - Simplify.
$1 > - 10$
$- 9 < 20$

Step 4 - True or false statement?
True
True
Since the ordered pair makes both inequalities true, it is a solution to the system of inequalities.

b. No.

Here is how to determine if each ordered pair is a solution:

Step 1 - Choose an ordered pair.
$(4, - 1)$

Step 2 - Substitute into the inequality.
$y > 4 x - 2$
$- 1 > 4 (4) - 2$

$4 x - y < 20$
$4 (4) - 1 (- 1) < 20$

Step 3 - Simplify.
$- 1 > 14$
$17 < 20$

Step 4 - True or false statement?
False
True
Since the ordered pair does not make a true statement in the first inequality, it is not a solution to the system of inequalities.