To solve these linear inequality problems, we need to graph each inequality on the coordinate plane. Here’s how to approach each one:
1. \( y \leq -\frac{1}{3}x + 2 \)
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Step 1: Graph the line \( y = -\frac{1}{3}x + 2 \). This is a straight line with a slope of \(-\frac{1}{3}\) and a y-intercept at (0, 2).
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Step 2: Since the inequality is \( \leq \), draw a solid line (because points on the line are included).
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Step 3: Shade the region below the line (since \( y \) is less than or equal to the expression).
2. \( y \leq \frac{1}{2}x - 5 \)
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Step 1: Graph the line \( y = \frac{1}{2}x - 5 \). This line has a slope of \(\frac{1}{2}\) and a y-intercept at (0, -5).
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Step 2: Draw a solid line because of the \( \leq \) sign.
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Step 3: Shade the region below the line.
3. \( y \geq \frac{1}{8}x + 2 \)
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Step 1: Graph the line \( y = \frac{1}{8}x + 2 \). The slope is \(\frac{1}{8}\) and the y-intercept is (0, 2).
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Step 2: Draw a solid line because of the \( \geq \) sign.
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Step 3: Shade the region above the line.
4. \( y > \frac{1}{4}x - 3 \)
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Step 1: Graph the line \( y = \frac{1}{4}x - 3 \). The slope is \(\frac{1}{4}\) and the y-intercept is (0, -3).
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Step 2: Draw a dashed line because of the \( > \) sign (points on the line are not included).
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Step 3: Shade the region above the line.
5. \( \frac{1}{6}x + \frac{1}{10}y \leq \frac{1}{2} \)
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Step 1: Rewrite in slope-intercept form:
\[
\frac{1}{10}y \leq -\frac{1}{6}x + \frac{1}{2}
\]
\[
y \leq -\frac{10}{6}x + 5
\]
\[
y \leq -\frac{5}{3}x + 5
\]
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Step 2: Graph the line \( y = -\frac{5}{3}x + 5 \). The slope is \(-\frac{5}{3}\) and the y-intercept is (0, 5).
-
Step 3: Draw a solid line because of the \( \leq \) sign.
-
Step 4: Shade the region below the line.
6. \( 5x \leq -4y - 12 \)
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Step 1: Rewrite in slope-intercept form:
\[
5x + 12 \leq -4y
\]
\[
y \leq -\frac{5}{4}x - 3
\]
-
Step 2: Graph the line \( y = -\frac{5}{4}x - 3 \). The slope is \(-\frac{5}{4}\) and the y-intercept is (0, -3).
-
Step 3: Draw a solid line because of the \( \leq \) sign.
-
Step 4: Shade the region below the line.
Final Answer:
For each inequality, follow the steps outlined above to graph the line and shade the appropriate region. Ensure to use solid lines for \( \leq \) or \( \geq \) and dashed lines for \( < \) or \( > \). Shade the correct side of the line based on the inequality sign.
Parent Tip: Review the logic above to help your child master the concept of worksheet on graphing inequalities.
