The garden and the deck together form a rectangle.
The sides of this rectangle are (15+x) and (20+x) feet. Recall that the area of a rectangle is the product of its length and width.
A=(15+x)(20+x)
Let's write and simplify an inequality representing that the total area cannot exceed 500 square feet.
We can use a calculator to sketch the graph of the quadratic corresponding to this inequality. Begin by pushing the Y= button and typing the equation in the first row.
To see the graph you will need to adjust the window. Push WINDOW, change the settings, and push GRAPH to see the graph.
Let's copy the graph from the calculator screen. Since x represents a distance, only positive values are meaningful. The solution of the inequality is the set of values of x for which the graph is not above the horizontal axis.
We see that the possible widths of the deck has an upper bound. To find this bound, we either use a calculator or solve a quadratic equation.
x^2+35x-200=0
From the calculator screen we can guess that the solution is x=5. Let's check.
The x-intercept of the graph is at x=5. We can combine this result with the constraint that x must be positive to write the solution set to the inequality.
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