Mr. Wheat is planning to take the Jazz Club to a music festival. Lawn tickets cost $20, and pavilion tickets cost $30. If he plans to spend at most $300, how many of each ticket can Mr. Wheat purchase?
Let
x represent the number of lawn tickets. Let y represent the number
of pavilion tickets. Then the inequality below represents the solution.
The inequality is
written in two variables. It is similar to an equation written in
two variables. An easy way to show the solution of an inequality is to
graph it in the coordinate plane. This
problem will be solved in Example 3.
The solution set
of an inequality in two variables contains many ordered pairs.
The graph of these ordered pairs fills an area on the coordinate plane
called a half-plane. The graph of an equation defines
the boundary or edge for each half-plane. Use these
steps to graph y >
3.
1.
Graph y
> –x + 3.
Alternative Solutions:
When graphing inequalities, the boundary line is not always dashed. Consider the graph of y ≥ 3. Since the inequality means y > 3 or y = 3, the boundary is part of the solution. This is indicated by graphing a solid line.
2.
Graph 4x
+ y ≤ 12.
Alternative Solutions:
To make a table or
graph for the boundary line, solve the inequality for y in terms of x.
When solving real-life inequalities, the domain and
range of the inequality are often restricted to nonnegative numbers or whole
numbers.
Budgeting Link
3. Refer to
the application at the beginning of the lesson. How many lawn and pavilion
tickets can Mr. Wheat purchase?
Alternative Solutions:
First, solve for y in
terms of x.
Mr. Wheat cannot buy a
negative number of tickets, nor can he buy portions of tickets. The solution is
positive ordered pairs that are whole numbers beneath or on the graph of the
line 
.
One solution is (12, 2).
This represents 12 lawn tickets and 2 pavilion tickets costing $300.
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