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Graph the line \( y = 2x + 5 \) as a dashed line. Shade the area below this line.
Solución
1. Begin with the inequality:
\[
-2x + y + 4 < 9
\]
2. Subtract \(4\) 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 given by the line:
\[
y = 2x + 5
\]
Since the inequality is strict (\(<\) and not \(\leq\)), the boundary line is dashed.
5. To determine the shading, use a test point. A common choice is \((0,0)\):
\[
0 < 2(0) + 5 \quad \Rightarrow \quad 0 < 5
\]
Since the test point satisfies the inequality, the region that contains \((0,0)\) should be shaded.
6. The final graph is a dashed line for \( y = 2x + 5 \) with the region below the line (where \( y < 2x + 5 \)) shaded.
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Simplifique esta solución Extra Insights
To graph the inequality \(-2x + y + 4 < 9\), first, we simplify it to isolate \(y\): \(-2x + y < 5\) or \(y < 2x + 5\). Next, you'll want to graph the line \(y = 2x + 5\). Start by plotting the y-intercept at (0, 5) and another point by substituting an x-value, like \(x = -2\) (which gives you \(y = 1\)), resulting in the point (-2, 1). Draw a dashed line through these points because the inequality is strict (<) and does not include equals. Lastly, shade the area below the line, indicating all the points where \(y\) is less than \(2x + 5\). This represents the solution set for the inequality.
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