Download Lesson 10 – Inequalities

Inequalities
Lesson 10
The warm up is on two step equations. We will review two step equations because in
this lesson you will solve inequalities a like you solve equations.
We are going to “S” our problem. Melissa and two friends are going to the movies.
The cost for admission for everyone is more than 21 dollars. What it she least that a
movie ticket may cost? In studying the problem we are going to underline the
question, what is the least that a movie ticket may cost? This problem is asking me
to find the least amount a movie ticket could cost.
When we solve the equation X minus 5 equals 12; we want to isolate the X. We have
a minus 5 in the way of the X; so we will add 5 to each side in order to isolate the X.
This leaves us with X equal to 17. If we want to check our answer, we will rewrite our
original equation of X minus 5 equals 12; we will plug 17 in for X; and 17 minus 5 is
12. If we solve the inequality X minus 5 is less than 12; we still want to isolate our X;
if we want to solve the inequality X minus 5 less than 12; we will solve this in the
same way we solved our equations. We have minus 5 so we are going to add 5 to
both sides; this leaves us with X is less than 12 plus 5; which is 17. We can choose
any number to check which is less than 17; if we chose the number 11 and plugged
it into our original equation. 11 minus 5 is equal to 6; and 6 is less than 12. So our
answer checks.
If we want to solve the problem 2X equals negative 8; we see there is a multiply by 2
on the same side as the X; in order to get rid of a multiply by 2; we are going to divide
each side by 2; this leaves us with X equal to negative 4. If we rewrite our original
equation of 2X equals negative 8; we can plug in our answer of negative 4 because 2
times negative 4 is negative 8; our answer is correct. If we look at the inequality 2X
is greater than or equal to negative 8, we will solve it in the same way; we divide both
sides by 2; and X is greater than or equal to negative 4. We can check our equation
by rewriting our original equation and plugging in an answer which is true. If X is
greater than or equal to negative 4; we can choose any number which is greater than
or equal to negative 4; negative 4 is equal to negative 4 so we can plug negative 4 in
for X. And negative 8 has to be equal to negative 8. This is a true statement so our
answer is correct.
We have the statement 8 is less than 10 which is a true statement. If we multiply
each side of the inequality by negative 2; we end up with negative 16 is less than
negative 20. This is not a true statement; if we want this to be a true statement; we
must invert or flip the inequality sign; because negative 16 is great than negative 20;
when we multiply a inequality by a negative we must invert or flip the inequality sign.
If we have the inequality 8 is less than 10; this is a true statement; if we divide both
sided of the inequality by negative 2; we get a statement of negative 4 is less than
negative 5; negative 4 is not less than negative 5; this is not true; in order for this to
stay true, we must invert the inequality sign or flip to change the inequality sign from
less than to greater than; now we have a true statement. When you divide an
inequality by a negative you must flip the inequality sign.
So solve the inequality negative 3X is less than 15; we have to isolate the X; in order
to isolate the X we have a multiply by negative 3; the opposite of multiply by negative
3 is to divide by negative 3; when we divide both sides by negative 3; negative 3
divided by negative 3 cancels and leaves us with X; when you divide an inequality by
a negative you must flip the inequality sign; from less than to greater than; 15 divided
by negative 3 is negative 5. If we want to check the inequality we rewrite the original
inequality and we choose a value of X which is true; we could say X is equal to
negative 4 because negative 4 great than negative 5; if we plug negative 4 in for X;
and negative 3 times negative 4 gives us 12; Is 12 less than 15; yes, so our answer
is correct.
To solve the inequality X divided by negative 2 plus 3 is greater than 9; we will first
isolate the X by subtracting 3 from both sides; when you subtract you do not ever
change the inequality sign it will always stay the same; and 9 minus 3 is 6; in order to
isolate the X we are dividing by negative 2 so we must multiply both sides by negative
2; the opposite of divide by negative 2 is multiply by negative 2; this leaves us with X
and because we multiplied by a negative we must flip the inequality sign; so X is less
than negative 12; If we want to choose which is less than negative 12 to check; we
could use a number such as negative 16 because negative 16 is less than negative
12; we will plug negative 16 into the inequality; negative 16 divided by negative 2 is
8; and 8 plus 3 is 11. 11 is greater than 9; so our answer is correct;
To solve the inequality we will first isolate the 2X by subtracting 7; the opposite of
plus 7 is subtract 7; this leaves us with 2X is greater than 2; we have 2 times X so the
opposite of multiply by 2 is to divide by 2; and X is greater than 1. In order to graph
this inequality we have greater than so we will use and open circle on one; and we
will draw and arrow to the right because the numbers to the right are greater than 1.
To solve the inequality negative 3X plus 4 is greater than or equal to 1; we are first
going to subtract 4 from both sides; so that we can isolate the negative 3X; the
opposite of plus 4 is minus 4; we have negative 3X is greater than or equal to
negative 3; in order to isolate our X we have a multiply by negative 3; the opposite of
multiply by negative 3 is divide by negative 3; our negative 3’s cancel; when you
divide by a negative you have to flip the inequality; greater than or equal to changes
to less than or equal to; negative 3 divided by negative 3 is equal to 1; our answer is
X is less than or equal to 1. To graph this inequality, because we have less than or
equal to we will put a closed circle on 1 and draw an arrow to the left; because the
numbers to the left are less than.
In order to complete our SOLVE problem, we already know that our question is what
is the least that a movie ticket may cost? If we “S” the problem, we found out that the
problem is asking me to find, the least amount a ticket could cost. In “O” we are
organizing the facts, we want to identify the facts. Melissa and 2 friends are going to
the movies, is our first fact; the cost for admission for everyone is more than 21
dollars, that’s our second fact. Both of those facts are necessary so we will list those
two facts. In “L” we are going to choose and operation and write in words what our
plan of action will be when we line up our plan. Choosing an operation we will use
division because we are trying to find the cost of each person; we are going to create
and solve an inequality with where the number of tickets; times the cost of one ticket,
T, is greater than the price given in the problem. If we want to estimate our answer in
verifying our plan with action; there are several different ways to estimate. Some
students may say that the cost has to be less than 21 dollars; some students may be
able to do the math in their head and say the cost may be less than 7; some students
may say that the cost will be around 5 dollars. We will now carry out our plan. We
said that the 3 tickets have to be greater than 21. We will divide both sides of the
inequality by 3; and see that the ticket cost has to be greater than 7 dollars; In “E”
examining our results we need to answer the question does your answer make
sense; our question was what is the least that a movie ticket may cost?; and we have
greater than 7 dollars; that answer does make sense. Is our answer reasonable? Our
estimate was more than 5 dollars so a ticket costing 7 dollars is a reasonable
answer; Is your answer accurate; We can choose a ticket price which is greater than
7; and check it in our inequality; 10 is greater than 7 so 30 is greater than 21; Our
tickets cost must be greater than 7 dollars; When we write our answer in a complete
sentence.
We are going to close our lessons with the essential questions. How do the answers
for equations differ from the answers for inequalities? And equation has one specific
answer where inequalities have several solutions. Number 2, when do you use an
open circle for graphing inequalities? And open circle is used when graphing a less
than or a greater than. Number 3, when do you use a closed circle for graphing
inequalities? A closed circle is used when graphing a less than or equal to or a
greater than or equal to.