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Algebra 1

2.14.2 Finding a Pair of Values that Satisfies Multiple Inequalities

Algebra 12.14.2 Finding a Pair of Values that Satisfies Multiple Inequalities

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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.

Close up of a quilt

Here are two graphs that represent the two constraints.

Length Cost
An inequality representing the length of fabric needed for a quilt is graphed on a coordinate plane. The x-axis represents the yards of light fabric while the y-axis represents the yards of dark fabric. The boundary line intersects the y-axis at 9.5 and extends to intersect the x-axis at 9.5. The region above the solid boundary line is shaded. An inequality representing the cost for the fabric needed to make a quilt is graphed on a coordinate plane. The x-axis represents the yards of light fabric while the y-axis represents the yards of dark fabric. The boundary line intersects the y-axis at 8 and extends to intersect the x-axis at 12. The region below the solid boundary line is shaded.
1.

Write an inequality to represent the length constraint. Let xx represent the yards of light fabric and yy represent the yards of dark fabric.

2.

Select three pairs that satisfy the length constraint.

  • ( 5 , 5 ) (5,5)
  • ( 2.5 , 4.5 ) (2.5,4.5)
  • ( 7.5 , 3.5 ) (7.5,3.5)
  • ( 12 , 10 ) (12,10)
3.

Write an inequality to represent the cost constraint.

4.

Select three the pairs that satisfy the cost constraint.

  • ( 4 , 5 ) (4,5)
  • ( 10 , 1 ) (10,1)
  • ( 8 , 3 ) (8,3)
  • ( 1 , 1 ) (1,1)
5.

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

6.

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

7.

What does the pair of numbers represent in this situation?

8.

Write the two inequalities together as a system of inequalities.

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?

    1. 20 seated tickets and 50 standing-room tickets
    2. 10 seated tickets and 60 standing-room tickets
    3. 40 seated tickets and 19 standing-room tickets
    4. 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.

    {x+4y103x2y<12{x+4y103x2y<12

    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)(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)(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:

    {x+4y103x2y<12{x+4y103x2y<12

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

    a.) Is the ordered pair (2,4)(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)(2,4) made both inequalities true. Therefore (2,4)(2,4) is a solution to this system.

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

    Try it

    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:

    {y>4x24xy<20{y>4x24xy<20

    a.)  (2,1)(2,1)   b.) (4,1)(4,1)

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