Graph the inequality:
$6(y+1)<-x$
( ____ , ____ )( ____ , ____ )
We are given the linear inequality \(6(y+1) < -x\). Our goal is to determine the nature of the boundary line, identify two points on this line, and determine the region that satisfies the inequality. The boundary line is defined by the equation \(6(y+1) = -x\). To simplify this, we can distribute the 6 on the left side:
\[6y + 6 = -x\]
Next, we isolate \(y\) to express the line in slope-intercept form \(y = mx + b\):
\[6y = -x - 6\]
\[y = -\frac{1}{6}x - 1\]
This form makes it easy to identify the slope \(m = -\frac{1}{6}\) and the y-intercept \(b = -1\).
The inequality given is \(6(y+1) < -x\), which is a strict inequality (using the less-than symbol \(<\)). Because the inequality is strict, the points exactly on the boundary line \(y = -\frac{1}{6}x - 1\) do not satisfy the condition. Therefore, the boundary line must be represented as a dashed line.
To find two points on the boundary line \(y = -\frac{1}{6}x - 1\), we can choose two arbitrary values for \(x\) and calculate the corresponding \(y\) values.
\[y = -\frac{1}{6}(0) - 1 = -1\]
So, one point on the line is \((0, -1)\).
\[y = -\frac{1}{6}(-6) - 1 = 1 - 1 = 0\]
So, another point on the line is \((-6, 0)\).
To determine which side of the boundary line satisfies the inequality \(6(y+1) < -x\), we test a point that is not on the line. The simplest point to test is the origin \((0, 0)\).
Substituting \((0, 0)\) into the inequality:
\[6(0+1) < -(0)\]
\[6(1) < 0\]
\[6 < 0\]
This statement is false. Since the origin does not satisfy the inequality, the solution region is the half-plane that does not contain the origin. Given the line \(y = -\frac{1}{6}x - 1\), the inequality \(y < -\frac{1}{6}x - 1\) represents the region below the line. Thus, the region to be shaded is the one below the dashed line.
Linear inequalities are widely used in resource allocation and optimization. For example, a budget constraint can be modeled as a linear inequality, such as \(10x + 5y \leq 100\), where \(x\) and \(y\) are quantities of two items. By graphing the inequality and identifying the shaded region (the feasible region), a business owner can visualize all combinations of goods they can afford to purchase while remaining within their budget. Understanding which side of a boundary line to shade is essential for finding the valid solutions to such practical problems.