Question
\( y \leq-\frac{3}{4} x \) Plot points on the boundary line. Select the line to switch betwee region to shade it.
Ask by Gonzalez Watson. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To graph \( y \leq -\frac{3}{4}x \), follow these steps:
1. **Draw the Boundary Line**:
- Plot the line \( y = -\frac{3}{4}x \) using points like (0,0), (4,-3), and (-4,3).
- Since the inequality includes “≤”, the line is solid.
2. **Determine the Shaded Region**:
- Choose a test point not on the line, such as (0, -1).
- Substitute into the inequality: –1 ≤ 0, which is true.
- Shade the region below the line, including the line itself.
3. **Finalize the Graph**:
- The shaded area represents all points (x, y) that satisfy \( y \leq -\frac{3}{4}x \).
Your graph will show a solid line with the region below it shaded.
Solution
We want to graph y ≤ –(3/4)x. Follow these steps:
1. Draw the boundary line:
• Replace "≤" with "=" to work with y = –(3/4)x.
• Pick a few x‐values to get points on the line:
– If x = 0, then y = –(3/4)(0) = 0 → (0, 0)
– If x = 4, then y = –(3/4)(4) = –3 → (4, –3)
– If x = –4, then y = –(3/4)(–4) = 3 → (–4, 3)
• Plot these points and draw the line through them.
• Because the inequality includes “=” (≤), the line is solid.
2. Decide which region to shade:
• Choose a test point that is not on the line (commonly (0, –1) works well).
• Substitute (0, –1) into the inequality:
– Left side: y = –1.
– Right side: –(3/4)(0) = 0.
– Check: –1 ≤ 0 ✓ (this is true).
• Since (0, –1) satisfies the inequality, the region containing (0, –1) is the solution set.
• In this case, that region is “below” (or on) the line.
3. Finalize the graph:
• With the boundary line in place and after confirming the test point works, shade the entire region below (and including) the line.
4. (Optional Instruction for Interactive Tools)
• If you are using an interactive graphing tool where you can “select” the line to toggle the shading option, click on the boundary line and choose “shade below” to display the solution region.
Your graph now shows a solid line passing through (0, 0), (4, –3), and (–4, 3) with the region below it (including the line) shaded to indicate all points (x, y) that satisfy y ≤ –(3/4)x.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
To graph the boundary line of the inequality \( y = -\frac{3}{4} x \), start by plotting two points. When \( x = 0 \), \( y = 0 \) gives the point (0,0). When \( x = 4 \), \( y = -3 \) gives the point (4,-3). Connect these points to form a line. Since the inequality is \( \leq \), this line will be solid, indicating that points on the line are included in the solution set. Next, shade the region below the line since it represents all points where \( y \) values are less than or equal to the values on the line. Remember, you can check a test point like (0, -1) to confirm that it lies below the line and satisfies the inequality! Happy graphing!
