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Answer:
3. D, Additive Property of Equality
4. x < 2
5. attached graph
Step-by-step explanation:
3. Additive Property of Equality: this means that you can add/subtract a number from one side, as long as you can do it to the other. as shown in steps 4 and 5, this is what they have done in order to get x by itself.
4. to solve -3x + 8 > 2x - 2, we want to get <em>x</em> by itself on one side. we can treat the inequality sign as an equal sign to solve
-3x + 8 > 2x - 2 < -- subtract 8 from both sides to get -3x alone
-8 -8
-3x > 2x - 10 < -- subtract 2x from both sides to get 2x with the -3x
-2x -2x
-5x > -10 <-- divide both sides by -5 to get <em>x</em> alone
when we divide by a negative number in an inequality, we flip the sign. so > becomes < once we divide by -5. this is important to ensure we get the right answer
-5x/-5 = x
-10/-5 = 2
x < 2
this inequality means that any value for <em>x</em> has to be <em>less than</em> 2 in order for the inequality to be true
on a number line it would look like the following:
<---------------2) - - - - - - - - >
the numbers to the left of 2 would be shaded in, as those are less than 2. the number 2 would have either an open circle on top of it or a parentheses facing to the left, either one works.
5. to graph y < 2/5x - 4, we want to get this equation into slope intercept form which is y = mx + b, where mx is the slope and b is the y-intercept
again, treat the inequality symbol as an = sign in this case and solve to get y by itself
y = 2/5x - 4 <--- this is already in slope intercept form, so we do not need to solve for y in this case.
to graph this, we would need to start at the y-intercept which is (0, -4). i will attach a graph to make this a bit easier visually
from there, you would use the slope to find another point on the graph. the slope is rise/run which also means y/x.
so you would need to go 2 units up from (0, -4), and 5 units right from where you landed
since this is an inequality, we will need to find any answers on the graph that satisfy the inequality by shading in the region where those points are solutions.
to do this we need to find a test point. a test point tells us whether or not that point and anything in its section is included in the solution. if the inequality is true with the test point, that answer and anything in that section is included in the answer. if its false, that point and everything in that section is not included.
you can always use the origin (0,0) if its not graphed on the line. in this case, we will use the origin as a test point. so plug in the test point into the original inequality
y < 2/5x - 4
0 < 2/5(0) - 4
0 < -4 <-- this is false and (0,0) and anything in its section is not included as a solution, so you would shade below the line as (0,0) was above the line, and anything above the line cannot be used as a solution to the inequality
graph the line connecting the two points using a dotted line as the inequality symbol is <, meaning points on that line are not solutions to the inequality
with the graph i attached, shade in the entire dark blue area. the graph is a bit rough but your graph should look similar to that
