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Which system of inequalities represents the graph?

AngelRay  Dec 19, 2017
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 #1
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Which system of inequalities represents the graph?

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Omi67  Dec 19, 2017
 #2
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Let's break this equation down step-by-step. Let's start with the solid line that goes upward from left to right first. The most useful type of linear equation is slope-intercept because it is easy to build the equation based on individual features of each linear equation.Let's pretend, for now, that the lines are not inequalities and instead equations.

 

\(y=mx+b\) Visually, the y-intercept is at (0,1), which allows us to plug y-coordinate of this corordinate in for b.
\(y=mx+1\) The slope can be calculated by observing rise over run here. In this case, the difference in the y-coordinates divided by the difference in the x-coordinates equals 2/7.
\(y=\frac{2}{7}x+1\) There are two problems with the equation we have generated thus far. One is that it is not an inequality. Secondly, it looks nothing like the given equations in the answer choices. First, let's deal with the correct inequality sign. Since the shaded region contains all the y-values below the original output, we know that we need a less-than-or-equal-to sign.
\(y\leq\frac{2}{7}x+1\) Now, we need to convert this into standard form so that we can compare this answer with the others provided. Eliminate the fraction since none of the answer choices contain a fraction anywhere.
\(7y\leq 2x+7\) Let's subtract 7 from both sides.
\(7y-7\leq 2x\)  
   

 

This immediately eliminates the last option since the inequality symbol is inverted, which in incorrect. Unfortunately, though, we can manipulate this answer and get the rest of the answers, so we will have to move on to the next line. Let's do the solid line that goes downward from left to right. Let's utilize the exact same process as before. 

 

\(y=mx+b\) The y-intercept appears to be located at the Cartesian coordinate (0,-8) and the slope is -2.
\(y=-2x-8\) Now, let's consider the inequality symbol. Every value on the line and less than what the line produced is shaded, so there needs to be a less-than-or-equal-to symbol for a second time instead of an equal sign.
\(y\leq -2x-8\) Multiply by 2 on both sides to eliminate the fraction, as it is difficult to compare the answer choices with the fraction remaining.
   

 

We can eliminate the first answer choice once again because the sign is incorrect. We can also eliminate the second option choice, too, because it is incorrect as well. It is 

 

\(-y\leq 2x+8\)

 

This is invalid because when multiplying or dividing by a negative number, the inequality sign is flipped, so this is incorrect, too. 

 

Therefore, there is no need to check for the other line, so the third option is correct.

TheXSquaredFactor  Dec 19, 2017

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