The answer is: [C]: "Graph C" . _________________________________________ Explanation: _________________________________________________
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. ____________________ Given the inequality: _________________ " <span>y - 3 > 2(x + 1) " ; ___________________ Let us simplify: " 2(x+1)" on the right-hand side of the inequality given: _____________________________________ Note the distributive property of multiplication: _______________________ a(b+c) = ab + ac ; AND: a(b - c) = ab - ac ; _______________________ So, 2(x+1) = 2*x + 2*1 = 2x + 2 ; _____________________________ Now, we can rewrite the inequality: ____________________________ </span> y - 3 > 2x + 2 ; _____________________________ Now, add "3" to EACH side of the inequality ; to isolate "y" on one side of the inequality: ______________________________ y - 3 + 3 > 2x + 2 + 3 ; ______________________________ to get: ______________________________ y > 2x + 5 ; _______________________________ Now, we can look at the graphs and identify more easily the correct answer. ________________ 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)". ______________________________ So, we have our choices narrowed down to "Graph A" and "Graph C". _________ 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: ______________ "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). __________________ So, let's start with: "Graph A". In the shaded area, select a random point within that shaded area. Let's choose, "(1,1)". ______________________________________ 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. ______________ 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". ______________________ Considering: " y > 2x + 5 ", let's plug the "x-coordinate", which is "-5", into:
"(2x +5)" and see what we get: _______________________________ 2*-5 + 5 = -10 + 5 = -5.
Is our "y-coordinate", which is "0", greater than "-5"??
In other words, is: 0 > -5 ?? YES! ______________________________________ So, the answer is: ______________________________________ Answer choice: [C]: "Graph C" . ________________________________________
To find a function none of the x-values can be the same. For example, #1 isn't a function because 2 of the x values re 2 meaning it is not a function. #4 is a function because all of the x values are different numbers.
