Unit 7, Lesson 9
Modeling with Systems of Inequalities in Two Variables
Integrated Math 1
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The organizer of a conference needs to prepare at least 200 notepads for the event and has a budget of \$160 for the notepads. A store sells notepads in packages of 24 and packages of 6.
This system of inequalities represents these constraints: \(\begin{cases} 24x+6y\geq200\\16x + 5.40y\leq160 \end{cases}\)
Here are incomplete graphs of the inequalities in the system, showing only the boundary lines of the solution regions.
Which graph represents the boundary line of the second inequality?
A system of inequalities graphed on a coordinate plane, origin O. Horizontal axis, number of large packages, scale from 0 to 12, by 2’s. Vertical axis, number of small packages, scale from 0 to 40, by 4’s. First line starts on vertical axis at approximately 33, goes through approximately 4 comma 18 and ends on the horizontal axis at about 8. Second line starts on vertical axis at approximately 29, goes through approximately 4 comma 18 and ends on the horizontal axis at about 10.
Complete the graphs to show the solution set to the system of inequalities.
A certain stylist charges \$15 for a haircut and \$30 for hair coloring. A haircut takes on average 30 minutes, while coloring takes 2 hours. The stylist works up to 8 hours in a day, and she needs to make a minimum of \$150 a day to pay for her expenses.
Graph the inequalities, and show the solution set.
Choose the graph that shows the solution to this system: \(\begin{cases} y > 3x + 2 \\ \text-4x+3y \leq12 \end{cases}\)
Graph of 2 intersecting lines, origin O. Scale -10 to 8 on both axes. Lines start in lower left and extend to upper right. Solid line passes through negative 6 comma negative 4 and 0 comma 4. Dotted line passes through negative 2 comma negative 4 and 0 comma 2. They intersect around 1 comma 6. The area to the right of the intersection point and below both lines is shaded.
Graph of 2 intersecting lines, origin O. Scale -10 to 8 on both axes. Lines start in lower left and extend to upper right. Solid line passes through negative 6 comma negative 4 and 0 comma 4. Dotted line passes through negative 2 comma negative 4 and 0 comma 2. They intersect around 1 comma 6. The area between the lines above their intersection point is shaded.
Two inequalities graphed on a coordinate plane, origin O, scale from negative 10 to 10 on both axes. The dashed line passes through negative 4 comma negative 10, 0 comma 2, and 2 comma 8. The area below the dashed line starting at the intersection point is shaded. The solid lines passes through negative 6 comma negative 4, 0 comma 4, and 3 comma 8. The area below the solid line until the intersection point is shaded.
Two inequalities graphed on a coordinate plane, origin O, scale from negative 10 to 10 on both axes. The dashed line passes through negative 4 comma negative 10, 0 comma 2, and 2 comma 8. The area below the dashed line is shaded. The solid lines passes through negative 6 comma negative 4, 0 comma 4, and 3 comma 8. The area above the solid line is shaded. The area between and above the point of intersection is not shaded.
Match each inequality to the graph of its solution.
Graph A
Graph B
Graph C
Graph D
Graph E
\(2x - 5y \geq 20\)
\(5x + 2y \geq 20\)
\(4x - 10y \leq 20\)
\(4x - 5y \geq 20\)
\(2x + 10y \leq 20\)