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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 boundary line is derived from the equality \(6(y+1) = -x\), which simplifies to \(y = -\frac{1}{6}x - 1\).
  • Because the inequality is strict (\(<\)), the boundary line must be dashed.
  • Two points on the boundary line are \((0, -1)\) and \((6, -2)\).
  • Testing the point \((0,0)\) results in a false statement, so the shaded region is the side of the line not containing the origin.

Explanation

  1. Analyzing the Inequality and Boundary Line

The given expression is a linear inequality in two variables: \(6(y+1) < -x\). This inequality defines a region (a half-plane) in the Cartesian plane. The boundary of this region is the line obtained by replacing the inequality symbol with an equality sign: \(6(y+1) = -x\). Since the inequality is strict (using the symbol \(<\)), the boundary line itself is not part of the solution set, which means it must be represented by a dashed line.

  1. Finding Points on the Boundary Line

To find points on the boundary line, we first rewrite the equation \(6(y+1) = -x\) into slope-intercept form (\(y = mx + b\)). Starting with \(6(y+1) = -x\), we divide both sides by 6 to get \(y+1 = -\frac{1}{6}x\), then subtract 1 from both sides to obtain \(y = -\frac{1}{6}x - 1\).

To plot this line, we choose two values for \(x\):

  • For \(x = 0\): \(y = -\frac{1}{6}(0) - 1 = -1\). So, one point is \((0, -1)\).
  • For \(x = 6\): \(y = -\frac{1}{6}(6) - 1 = -1 - 1 = -2\). So, another point is \((6, -2)\).
  1. Determining the Shaded Region

To determine which side of the boundary line represents the solution, we choose a test point that is not on the line. The origin \((0, 0)\) is a convenient choice. We substitute \(x = 0\) and \(y = 0\) into the original inequality \(6(y+1) < -x\):
\(6(0+1) < -(0)\)
\(6(1) < 0\)
\(6 < 0\)
This statement is false. Therefore, the region containing the origin is not the solution set. The shaded region must be on the side of the boundary line that does not contain the origin. Assuming regions A and B are defined relative to this line, we select the region that does not include the origin.

Examples

In economics, inequalities like this are frequently used to define budget constraints. For example, if a consumer has a limited amount of money to spend on two products, the combination of quantities they can afford falls within a specific half-plane. Visualizing these constraints as regions helps in understanding the set of all affordable choices and making optimal purchasing decisions.

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