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Question 549968: How do I solve AND graph this conjunction?
-4y + 17 less than or equal to 29
Found 2 solutions by Theo, KMST:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! equation to solve is:
-4y + 17 <= 29
subtract 17 from both sides of the equation to get:
-4y <= 29 - 17
simplify to get:
-4y <= 12
divide both sides by -4 to get:
y >= -3
dividing both sides of an inequality by a negative number reverses the inequality.
to graph this solution, you would make the inequality an equality to create the equation of y = -3
you would then graph that equation.
you would then shade the area of the graph that corresponds to the original equation of y >= -3
your graph would look like this:
you graphed the equation of y = -3 and your solution set lies on or above that line so you shade the area over the graph of the equation of y = -3.
this means that any value of y >= -3 is a solution to your original equation.
since the solution is y >= -3, then the graph of the equation uses a solid line.
if the solution was y > 3, then the graph of the equation would use a dotted line to indicate that the solution set is above the line but not also on it.
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! There is a conjunction, but I would classify the problem as an inequality and would write it as
To graph inequalities with just one letter variable, you only need a number line. I'll describe the style of graphing that's most common in the USA. (If a different style is used in your class, you would know).
If "showing your work" is not required (as for multiple choice tests), you can use shortcuts to work faster. In any case, the same type of thinking helps you understand the problem better, and avoid mistakes.
For your inequality, you know that the y value that complies with  is one of the solutions (because of that "or equal"). So you find that y value and mark it as a filled in/colored in little circle on your number line.
Starting with , we could subtract 17 from both sides to get
 and then simplify to get
 . At this point, we would divide both sides by -4 to get
 --->  .
If you are not required to show more work, the rest is simple and fast. For inequalities as simple as yours, the solution will include either all the points to the left, or all the points to the right. To decide, you could use another point as a test point. Choose one that makes the calculations easy. If possible, I use zero. Is  part of your solution?
For ,  Since  , is a solution, and so are all the points on the same side of -3. So you graph your solution by adding an arrow starting from -3, and covering all your solutions, like this:
The solution is -3 and all the points to the right,  .
That blue arrow should be colored right on top of the number line, but I could not draw it any better.
If you need to show work, a little algebra work would be expected. It could look like this
Subtracting 17 from both sides would give you an equivalent inequality with the same sign, so we keep working with
 which simplifies to 
Dividing by 4 would give you the equivalent inequality
That minus sign, is the problem. Multiplying (or dividing) both sides of an inequality by a negative number changes the inequality.
If you multiply by -1 both sides of the inequality to change minus to plus (and plus to minus), you also have to flip the sign.
---> 
The flipping happens because multiplying everything times -1 is flipping the line with all the points on it (leaving in the same place only the zero point). You go from
 to
As you flip, everything flips: point positions, plus and minus signs, lesser than and greater than symbols. Once you changed all the minuses to pluses and all the pluses to minuses, the points for numbers less than 3, that were to the left of 3, ended up to the right of -3.
GENERAL NOTES:
The expression  is very simple. It involves only terms that are constant (like +17) and terms that are your one and only variable multiplied by a number. The variable is not involved in anything more complex than that: no exponents, no square roots, no variable as part of a denominator. If only that kind of simple expression is involved in an inequality, there will be one boundary point, the numbers to one side will be solutions, and the numbers to the other side will not be solutions. The boundary point will be a solution if the inequality involves an "or equal". If an "or equal" is not there, the boundary point is not part of the solution, and is shown in the graph as a little circle around the number, with the inside not filled in/colored in.
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