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Chapter 7: Problem 6

Graph each linear inequality. \(2 x-5 y<10\)

Short Answer

Expert verified
The graph of the inequality \(2x - 5y < 10\) is a dashed line given by \(y = \frac{2}{5}x - 2\), with the region below this line shaded.

Step by step solution

01

Turn the inequality into slope-intercept form

Rearrange the inequality \(2x - 5y < 10\) to get it into the standard form \(y = mx + b\). This is done by first subtracting \(2x\) from both sides, getting \(-5y < -2x + 10\). Then, divide everything by \(-5\) (and remember that when you divide by a negative number, the sign flips). You should get \(y > \frac{2}{5}x - 2\).
02

Draw the line

The line to be graphed is \(y = \frac{2}{5}x - 2\). This line intercepts the y-axis at \(-2\), and has a slope of \(\frac{2}{5}\), which means for every 5 units we move to the right, we move 2 units up. Remember to make this line dashed, because points right on the line do not satisfy the inequality (we have a strict 'greater than' inequality, not a 'greater than or equal to').
03

Choose which side to shade

In order to decide which side of the line to shade, we can pick a test point. The easiest point to pick is often the origin (0,0), unless the line passes through that point. In this case it doesn't, so plug (0,0) into the inequality \(y > \frac{2}{5}x - 2\). This gives \(0 > -2\), which is true, so the side that contains the origin is the solution, meaning that is the side to be shaded.

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