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Answer
Graph the line \( y = 2x + 5 \) as a dashed line. Shade the area below this line to represent the solution to the inequality \( -2x + y + 4 < 9 \).
Solution
1. Start with the inequality:
\[
-2x + y + 4 < 9
\]
2. Isolate \( y \) by subtracting 4 from both sides:
\[
-2x + y < 5
\]
3. Add \( 2x \) to both sides to solve for \( y \):
\[
y < 2x + 5
\]
4. The boundary of the inequality is the line:
\[
y = 2x + 5
\]
Since the inequality is strict (\( < \)), the line is drawn as a dashed line.
5. To determine which side of the line to shade, test a point not lying on the line. A common choice is the origin \((0, 0)\):
\[
0 < 2(0) + 5 \quad \text{which simplifies to} \quad 0 < 5
\]
This is true, so the region containing the origin satisfies the inequality.
6. Graphically, draw a dashed line for \( y = 2x + 5 \) and shade the region below it, since \( y \) is less than \( 2x + 5 \).
The resulting graph on the coordinate plane represents the solution to the inequality \( -2x + y + 4 < 9 \).
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Simplify this solution Beyond the Answer
To graph the inequality \( -2x + y + 4 < 9 \), first simplify it to \( y < 2x + 5 \). Start by graphing the line \( y = 2x + 5 \). This is a straight line with a slope of 2 and a y-intercept at 5. Remember to use a dashed line since the inequality is "<" and not "≤", indicating that points on the line are not included in the solution. Now, choose a test point not on the line, like (0,0). Plugging it into the inequality \( 0 < 0 + 5 \) yields true—this means you shade below the line, which represents all the solutions to the inequality! And there you have it, a fun-looking coordinate plane with your shaded region!
