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Chapter 7: Problem 5

Graph each linear inequality. \(2 x+3 y>12\)

Short Answer

Expert verified
The line \(y = \frac{-2}{3}x + 4\) forms the boundary for the inequality. The region above this line (not including it since the inequality is strictly greater-than) represents the solution to the inequality \(2x + 3y > 12\).

Step by step solution

01

Convert the inequality to the standard form

Begin by re-arranging \(2x + 3y > 12\) into the standard form for a linear equation, i.e., \(y = mx + c\). Here, we isolate y: \(3y > 12 - 2x\) divide by 3 to the both side to achieve the standard form: \(y > \frac{-2}{3}x + 4\)
02

Graph the line

Plot the line \(y = \frac{-2}{3}x + 4\). This line intercepts the y-axis at y = 4 and has a slope of \(-\frac{2}{3}\), so for every move of 3 units to the right, the line moves 2 units down.
03

Select the shaded region

Since the inequality symbol is '>', the region above the line will be shaded. If the inequality was '\(\ge\)', then the line would also be included in the solution. Here, the line isn't included, it simply forms a boundary so it should be drawn as a dashed line.

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