Answered

Graph the inequality:

$6(y+1) < -x$


  1. Choose the type of boundary line:
  • Solid
  • Dashed


  1. Enter two points on the boundary line:

( ____ , ____ ) ( ____ , ____ )


  1. Select the region you wish to be shaded:
  • Region A
  • Region B

Verified Answer :

  • The inequality \(6(y+1) < -x\) simplifies to \(y < -\frac{1}{6}x - 1\).
  • The boundary line is a dashed line because the inequality is strict.
  • Two points on the boundary line are \((0, -1)\) and \((-6, 0)\).
  • Testing the point \((0,0)\) shows it is not part of the solution, so the region below the line is shaded.
  • \(\boxed{\text{Boundary line: Dashed, Points: (0, -1) and (-6, 0), Region: B}}\)

Explanation

  1. Analyzing the Boundary Line

To determine the characteristics of the inequality \(6(y+1) < -x\), we first identify the boundary line by treating the inequality as an equality:
\[6(y+1) = -x\]
Expanding the left side, we get:
\[6y + 6 = -x\]
Solving for \(y\), we obtain the slope-intercept form of the line:
\[6y = -x - 6\]
\[y = -\frac{1}{6}x - 1\]
Since the original inequality is a strict inequality (\(<\)), the boundary line must be dashed.

  1. Identifying Points on the Line

We find two distinct points on the boundary line \(y = -\frac{1}{6}x - 1\).
First, let \(x = 0\):
\[y = -\frac{1}{6}(0) - 1 = -1\]
So, one point on the line is \((0, -1)\).
Next, let \(x = -6\):
\[y = -\frac{1}{6}(-6) - 1 = 1 - 1 = 0\]
So, another point on the line is \((-6, 0)\).

  1. Determining the Shaded Region

To identify the correct region to shade, we test a point that is not on the boundary line. Let's test the origin \((0, 0)\):
\[6(0+1) < -0\]
\[6 < 0\]
Since the statement \(6 < 0\) is false, the origin \((0, 0)\) is not in the shaded region. Because the origin lies above the boundary line, the shaded region must be the side that does not include the origin, which is the region below the line. Assuming region B refers to the area below the line, we choose region B.

Examples

In urban planning, inequalities help define restricted zones. If a local ordinance states that for a new development, the ratio of green space to building footprint must satisfy a certain linear inequality, engineers must calculate the boundary line of this region to ensure their building design complies with local codes. By graphing this boundary, they can visually identify the permissible zone for construction.

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