To graph the inequality \(8x + 5y \leq 100\), we first need to convert it to the slope-intercept form, which is easier to graph.
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Start with the inequality: \[ 8x + 5y \leq 100 \]
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Solve for \(y\): \[ 5y \leq -8x + 100 \] \[ y \leq -\frac{8}{5}x + 20 \]
Now we can identify the y-intercept and the slope:
- The y-intercept is \(20\) (this is where the line crosses the y-axis).
- The slope is \(-\frac{8}{5}\).
Steps to Graph the Inequality:
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Plot the y-intercept: Start by plotting the point (0, 20) on the graph.
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Use the slope: From (0, 20), move down 8 units and right 5 units to find another point on the line. This gives the point (5, 12).
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Draw the line: Since the inequality is \(\leq\), we will draw a solid line through the points (0, 20) and (5, 12) because the line itself is included in the solution.
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Shade the region: Since we want to represent \(y \leq -\frac{8}{5}x + 20\), we shade below the line. This indicates all the combinations of \(x\) and \(y\) that satisfy the inequality.
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Check intercepts: To find the x-intercept, set \(y = 0\): \[ 8x + 5(0) = 100 \implies 8x = 100 \implies x = 12.5 \] So, the x-intercept is (12.5, 0).
The solution set includes all combinations of \(x\) (bath towels) and \(y\) (hand towels) that satisfy this graph, bounded by the axes since the customer can’t buy negative quantities of towels.
Summary
Look for a graph that:
- Shows a solid line connect points (0, 20) and (12.5, 0).
- Shades the area below this line.
- The axes are positive (x ≥ 0 and y ≥ 0) since negative quantities don't make sense in this context.
These aspects are crucial to correctly identifying the solution set of the inequality.
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