The answer is: [C]: "Graph C" .
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Explanation:
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To make matters easier, we can "simplify" the inequality given, by rewriting in a format that isolates "y" on one side of the equation; similar to "slope-intercept form", which is " y = mx +b" ; only with an "inequality" sign replacing the "equals" sign.
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Given the inequality:
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" y - 3 > 2(x + 1) " ;
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Let us simplify: " 2(x+1)" on the right-hand side of the inequality given:
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Note the distributive property of multiplication:
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a(b+c) = ab + ac ; AND:
a(b - c) = ab - ac ;
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So, 2(x+1) = 2*x + 2*1 = 2x + 2 ;
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Now, we can rewrite the inequality:
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y - 3 > 2x + 2 ;
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Now, add "3" to EACH side of the inequality ; to isolate "y" on one side of the inequality:
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y - 3 + 3 > 2x + 2 + 3 ;
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to get:
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y > 2x + 5 ;
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Now, we can look at the graphs and identify more easily the correct answer.
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Looking at the graphs, and considering that our "inequality" is "y" is GREATER than "(2x+5)", we look at the answer choices given, and see that only "Graph A" and "Graph C" have a "dotted line"; so we can narrow our answer choices down to these, especially when we see the "(0,5)" which is a value when "y" EQUALS "(2x+5)".
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So, we have our choices narrowed down to "Graph A" and "Graph C".
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Now, let us look at the shaded area for EACH of these TWO graphs. Then find ANY point that lies WITHIN the shaded area, and plug the "x" coordinate" into the right side of the inequality:
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"y > 2x + 5 " ; That is, plug in that value for "x" in "2x + 5" and see what the value is, and see if that value is LESS THAN the "y" value of that (chosen point).
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So, let's start with: "Graph A". In the shaded area, select a random point within that shaded area. Let's choose, "(1,1)".
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Considering: " y > 2x + 5 ", let's plug the "x-coordinate", which is "1", into:
"(2x +5)" and see what we get: (2*1) + 5 = 2 + 5 = 7. So, is it true that the "y-coordinate" for our point "(1,1)", which is "1", is greater than "7"?
In other words, 1 > 7 ?? No! So, we know that "Graph A" is not the correct graph.
We are left with: "Graph C", which is answer choice: "C", which is the correct answer.
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But let's try a point in the shaded area of "Graph C".
Let's choose, say, "(-5,0)". The "x-coordinate" of this point is: "-5"; and the "y-coordinate" of this point is "0".
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Considering: " y > 2x + 5 ", let's plug the "x-coordinate", which is "-5", into:
"(2x +5)" and see what we get:
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2*-5 + 5 = -10 + 5 = -5.
Is our "y-coordinate", which is "0", greater than "-5"??
In other words, is: 0 > -5 ?? YES!
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So, the answer is:
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Answer choice: [C]: "Graph C" .
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