TEK 6.9 A/B, TEK 6.10 A/B - Inequalities
TEK 6.9 A Learning Goal: I can write one-variable, one-step equations and inequalities to represent constraints or conditions within problems.
TEK 6.9 B Learning Goal: I can represent solutions for one-variable, one-step equations and inequalities on number lines.
TEK 6.10 A Learning Goal: I can model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.
TEK 6.10 B Learning Goal: I can determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.
Note: We are focusing mainly on the inequality parts of these TEKS because that is what gives students the most trouble.
Vocabulary
Inequality: A statement that compares two quantities using <, >, ≤, or ≥.
Variable: A symbol (usually a letter) used to represent a quantity that can change, or an unknown number.
Constant: A value that does not change.
Solution: A value that makes an equation or inequality true.
Inequalities
Most math problems have one answer. For example, 4 times 2 is 8, and only 8, not any other number. But some problems will have a range of many different numbers that can be represented by an inequality, a comparison of two quantities.
When we say "comparison," we mean comparing the two quantities using these symbols: <, >, ≤, and ≥. Each of these symbols represents words.
< is short for "is less than."
> is short for "is greater than."
≤ is short for "is less than or equal to."
≥ is short for "is greater than or equal to."
Make sure that you memorize these symbols. They're very important for understanding inequalities. You can try writing sentences with these symbols. Look at this sentence:
6 < 7.
It doesn't look like a sentence, but when you put "is less than" instead of <, you get "6 is less than 7." That's a true statement.
8 ≥ 3.
When you put "is greater than or equal to" instead of ≥, you get "8 is greater than or equal to 3," which means that 8 could be equal to 3 or any number above it. 8 is greater than 3, so this statement is true. This statement is also true:
2 ≤ 2.
The statement reads "2 is less than or equal to 2." 2 must be either equal to 2 or less than 2. It is equal to 2, so the statement is true.
In most real-world problems, one of the quantities, or numbers, in the inequality will be represented by a variable, showing that it can change values. That makes sense - if inequalities are used to show answers that include many different numbers, of course one of the quantities has to be able to change to different numbers.
Greater Than
Let's say there is an amusement park with go-karts, and you have to be over 9 years old to ride the go-karts. If we were to ask what age we have to be to ride the go-karts, there is no single-number answer. You can be 10, but you can also be 11, and 12, and so on, as long as you are over 9 years old.
We would want to represent this situation with an inequality. Inequalities compare two quantities, one of which is usually a variable, which stands in place of a changing number. The other is the constraint that cuts off the possible answers.
Here, 9 years old is that constraint that cuts off possible answers, and your actual age, which can be represented by the variable a, is the other quantity in the inequality. To make the inequality, first try putting it into words:
If I am allowed ride the go-karts, my age (a) must be greater than 9.
Now replace "greater than" with the symbol >, because, if you look at the definition of each inequality symbol up above, > means "is greater than." Now you have:
If I am allowed to ride the go-karts, a > 9.
a > 9 represents all of the possible ages that you can be if you are allowed to ride the go-karts. Any number that you put in place of a has to make the inequality a true statement, or else that is not a valid age.
If you were 8 years old, you would put 8 in place of a, so that 8 > 9. This creates a problem because the statement "8 is greater than 9" is not true. So, you can't be 8.
But you can be 10, because if you put 10 in, you get 10 > 9. "10 is greater than 9" is a true statement, so you will be allowed to ride the go-karts if you are 10.
What if you are 9 years old? Let's put 9 in place of a. We get 9 > 9, or "9 is greater than 9." This statement is actually not true. 9 is equal to 9, but it is not greater than 9. If you are 9 years old, you just missed the cut-off and you will not be able to ride the go-karts.
Inequalities can be represented using a number line. Here is the number line for the go-kart problem:
As you can see, part of the line is highlighted in blue. Any number on the line that is directly under the blue highlighted part is a solution, or one of the ages you can be if you are allowed to ride the go-carts.
10 is under the blue highlight, so 10 is an age you can be if you want to ride the go-karts. 8 is under the regular, black part of the line, so you can't be 8 and ride the go-karts. But what's happening with 9?
On an inequality number line, the circle represents the threshold, or barrier between the solutions and the other numbers that are not solutions. The barrier is 9 because that was the constraint number given to us in the problem. But 9 can't be part of the solution because 9 is not greater than 9.
So, we leave the circle open, or unfilled. This shows that 9 is located under a spot on the number line that is not highlighted (look at how the number line is black directly above 9.) So, 9 is not included in the solutions part of the number line.
An open dot above a number means that it is not a solution.
Less Than
The main difference between a pony and a horse is that a pony has to be less than 14.2 hands tall. A hand is an ancient measurement that is now standardized at 4 inches.
Let's write the inequality that shows how tall the animal can be to be considered a pony. 14.2 is our constraint, or our threshold, so we know that 14.2 is one of the quantities we will be comparing. The other quantity will be a variable that can represent all the possible solutions, which is all the possible heights, in this case. Let's say that h is the variable that will represent height. Start with a sentence:
If that animal is a pony, then its height (h) it must be less than 14.2 hands tall.
Replace "less than" with <.
If that animal is a pony, then h < 14.2.
h < 14.2 is your inequality.
If your animal is 13.4 hands tall, then you would plug 13.4 into h, the height variable. 13.4 < 14.2 is a valid statement, so 13.4 hands tall is a pony.
If your animal is 15 hands tall, the inequality would be 15 < 14.2, which is not a true statement. So, 15 hands tall is not a pony, but a horse.
If the animal was exactly 14.2 hands tall, the inequality would change to 14.2 < 14.2. This isn't true either because 14.2 is equal to 14.2, not less than. So, an animal at 14.2 hands would not be a pony.
The number line for this inequality could look like this:
This time, the blue portion of the number line, which represents the solutions, extends (goes on and on) in the other direction, to the left, to show that numbers with a lower value than 14.2 are the solutions.
Again, there is an open dot at 14.2 because 14.2 is a threshold but is not included in the solutions because 14.2 < 14.2 is not true.
Greater Than Or Equal To
In Texas and most other states, you have to be at least 16 years old to have a driver's license. "At least" is a different phrase than "over" or "greater than," because it shows that you can not only be more than 16 and have a license, but you can also be 16 and have a license.
Let's write the inequality for people who are able to get their license. The threshold is 16 and the variable will be a to represent age.
If you have a driver's license, your age (a) must be greater than or equal to 16 years.
Change "greater than or equal to" into the corresponding symbol, ≥.
If you have a driver's license, a ≥ 16.
a ≥ 16 is our inequality.
Plug 13 years old in and see what you get. You should get an inequality that is not true, so you can't be 13 and have a driver's license.
Now try 20. You should get an inequality that is a true statement, so you can be 20 and have a driver's license.
What about 16? If you plug 16 in to a, you get 16 ≥ 16. The ≥ means "greater than or equal to," so the numbers being compared can be equal to each other, or the number on the left can be greater than the number on the right. 16 is equal to 16, so this inequality is true. So, 16 is a solution, and you can be 16 and have a driver's license.
Here's the number line:
This number line is almost exactly like the number line for the Greater Than example, except the dot on the threshold (16) is filled in. The number line above 16 is blue, not black, so this filled in dot shows that 16 is included as part of the solution.
Less Than Or Equal To
The speed limit for a residential street is 25 mph. This means you can travel at a speed of no more than 25 mph. "No more than" is different from "less than" or "under" because it shows that 25 is include in the solutions of possible speeds you can go. You can't ever travel above 25 mph, but you can travel at 25 mph and anywhere below.
Let's write the inequality for possible speeds you can go on this street without breaking the law. The threshold is 25 mph and the variable s represents possible speeds.
If you are following the law, your speed (s) must be less than or equal to 25 mph.
Replace "less than or equal to" with the corresponding symbol, ≤, and you have:
If you are following the law, s ≤ 25.
s ≤ 25 is your inequality.
Try some speeds above and below 25 mph, and see which ones satisfy the inequality (make it true.) All speeds above 25 mph should make the inequality untrue, so you are breaking the law if you are going that fast. All speeds below 25 mph should make the inequality true, so you are following the law at those speeds.
Now try 25 in place of s. The inequality becomes 25 ≤ 25, or "25 is less than or equal to 25." The two numbers being compared could be equal to each other, or the number on the left could be less than the number on the right. 25 is equal to 25, so the inequality is true. So, you can go 25 mph without breaking the law.
Here is the number line:
Again, there is a filled in dot at the threshold, 25, because 25 is part of the solution. The solutions extend to the left because numbers of lower value than 25 are also included as solutions.
Summary
For an inequality, write the variable that represents all possible solutions on the left, and write the threshold number that is given to you in the problem on the right.
If a problem includes the words "over," "greater than," "more than," "above," etc., then the inequality symbol you need is the greater than sign, or >.
If a problem includes the words "under," "less than," "lower than," "below," etc., then the inequality symbol you need is the less than sign, or <.
If a problem includes the words "at least," "greater than or equal to," etc., then the inequality symbol you need is the greater than or equal to sign, or ≥.
If a problem includes the words "no more than," "at most," "less than or equal to," etc., then the inequality symbol you need is the less than or equal to sign, or ≤.
If the situation is a "greater than" situation, the solutions on the number line will start at the threshold on an open dot, and extend to the right.
If the situation is a "less than" situation, the solutions on the number line will start at the threshold on an open dot, and extend to the left.
If the situation is a "greater than or equal to" situation, the solutions on the number line will start at the threshold on an closed dot, and extend to the right.
If the situation is a "less than or equal to" situation, the solutions on the number line will start at the threshold on an closed dot, and extend to the left.
Whenever you have a ≥ or ≤ situation, the threshold number, or the constraining number, is part of the solutions.
Solving Inequalities
Up until now, all of the problems we have done have a variable on the left and some constant (unchanging) number on the right. However, sometimes the variable side may also have a constant, like these inequalities:
x + 4 < 6
2x ≥ 16
4 and 2 in these inequalities are both constants. The constant and the variable (x) could be added, subtracted, multiplied, or divided by each other, depending on the problem. More on that later.
When we say "solving inequalities," we mean changing the appearance of the inequality so that the variable (x in this case) is by itself. To do this, we first need to learn some rules of inequalities.
Addition and Subtraction Properties
When you add or subtract the same number from each side of an inequality, the inequality remains true.
If we have this inequality:
9 > 6
We can subtract or add whatever number we want, as long as we do it from both sides. Try adding 2 to both sides.
The new inequality is 11 > 8. 11 is a higher number than 8, so the inequality is still true. Now try subtracting 2 from both sides of the original inequality, 9 > 6.
The new inequality is 7 > 4. 7 is a higher number than 4, so the inequality is still true.
The inequality will still be true even if you subtract a number big enough to turn 9 and 6 into negative numbers. Try subtracting 10 from both sides of the inequality:
Your new inequality is -1 > -4. Even though 1 is not greater than 4, -1 is certainly greater than -4, because on a number line -1 would be further to the right, closer to zero. So, the inequality remains true.
You can add or subtract any number you want, no matter how big or small or whether it is positive or negative.
Multiplication and Division Properties
Multiplication and division properties are similar to addition and subtraction properties. In fact, the rule is identical to the addition and subtraction rule when you are multiplying and dividing by a positive number. But when you are using a negative number, things start to get different. Here are the rules:
- When you multiply or divide each side of an inequality by the same positive number, the inequality remains true.
- When you multiply or divide each side of an inequality by the same negative number, the inequality symbol must be reversed for the inequality to remain true.
- When you reverse an inequality symbol, flip the < or > part of the symbol. If there is a line under these symbols, just leave it there.
- < becomes >
- > becomes <
- ≤ becomes ≥
- ≥ becomes ≤
Let's explore these rules with this inequality:
6 < 8
Try multiplying both sides by 2:
The new inequality is 12 < 16. 12 is lower than 16, so the inequality is still true. The inequality is still true because we were multiplying both sides by the same positive number, 2. Now try dividing both sides of 6 < 8 by 2:
The new inequality is 3 < 4, which is still a correct statement because 3 has a lower value than 4. Once again, the inequality stayed true because we divided by a positive number.
Let's try multiplication of both sides of 6 < 8 by -2:
The new inequality is -12 < -16. This isn't correct. -12 should be greater than -16 because it is further to the right on a number line, closer to zero.
Multiplying by a negative changed the signs. When you change a number's sign, it flips to the other side of the number line.
If both numbers were positive, then the one with the larger absolute value is the highest. But when two numbers are negative, the one with the lowest absolute value (the one closest to zero) is highest.
So, whenever you flip the sign on the numbers of an inequality, the symbol needs to flip as well. < changes to >, so the new inequality would be:
-12 > -16.
The same thing happens when you divide by a negative. Take this inequality:
9 > -3
Try dividing both sides by -3.
You get -3 > 1, which is incorrect. You need to flip the sign because you divided by a negative, and that would give you -3 < 1.
Notice how it doesn't matter what the original signs of the inequality are. They may both be positive numbers, or both negative, or one positive and one negative. If one number is positive and the other is negative, they get flipped to opposite sides of zero on the number line. If both numbers are positive or both are negative, flipping the signs changes whether having a higher absolute value makes a number higher or lower.
We already saw a number line for numbers that have the same sign, so now look at this example of two numbers with opposite signs after being multiplied by -1. We will multiply both sides of 5 > -4 by -1.
Solving
Now you have all the tools that you need to solve inequalities. Remember, the goal of solving inequalities is to get the variable (represented by x in the following problems) by itself on one side of the inequality symbol.
Solving Using Addition
You will need to use the addition property if a constant is being subtracted from the variable. One side of the inequality will look like this: x - a, where x is a variable and a is a constant.
Consider this inequality:
Since you are subtracting a constant from a variable, you will need to use the addition property. You need to find out what number you can add to both sides that will leave x by itself. If you add the number that you are subtracting from x (6), the addition and subtraction of the same number will cancel each other out. If you add 6 to both sides:
x > 19 is your final answer.
Solving Using Subtraction
You will need to use the subtraction property if a constant is being added to the variable. One side of the inequality will look like this: x + a, where x is a variable and a is a constant.
Consider this situation:
When a number represented by the variable x is added to 5, the sum is greater than or equal to 8.
First, turn this sentence into an inequality. The two sides of the inequality will be separated by the inequality symbol. The words "greater than or equal to" in the problem are equivalent to the symbol ≥. Now the sentence looks more like this:
x added to 5 ≥ 8.
You can put a plus sign to represent adding, which gives you:
x + 5 ≥ 8
Now you are ready to solve. Since you are adding a constant to x, you need to use the subtraction property. You need to find out what number you can subtract from both sides that will leave x by itself. If you subtract the same number you are adding to x (5), the subtraction and addition of the same number will cancel each other out. If you subtract 5 from both sides:
x ≥ 3 is your final answer.
Solving Using Multiplication
You will need to use the multiplication property whenever a variable is divided by a constant. One side of the inequality will look like this: x ÷ a (or x/a), where x is a variable and a is a constant.
Consider this inequality:
x ÷ (-8) < 3
Since you are dividing the variable by a constant, you need to use the multiplication property. You need to find out what number you can multiply both sides by that will leave x by itself. If you multiply by the same number you are dividing the variable by, the multiplication and division by the same number will cancel out to 1, leaving 1x, or just x, alone on the left side. Multiply both sides by -8:
x > -24 is your final answer. Don't forget to flip the signs!
Solving Using Division
You will need to use the division property whenever a constant is multiplied by the variable. One side of the inequality will look like this: ax, where x is a variable and a is a constant.
Consider this situation:
Laverne is making bags of party favors for each of the 7 friends attending her birthday party. She does not want to spend more than $42 total on party favors. Find an inequality that shows the maximum cost to make each bag.
This problem has a lot going on with it. First of all, we know that Laverne is trying to buy some party favors for her birthday. She wants to spend no more than $42 on these favors, so the total amount she ends up spending has to be less than or equal to $42.
Total amount ≤ 42
We don't have an exact number of how much each bag costs - it could be several different numbers. So, the cost of each bag is the variable. We will represent it with x.
If there are 7 friends, then Laverne will need to make 7 bags, each costing x dollars to make. The final price of all the bags will be 7 times x, or 7x. Now put 7x into our inequality in the "total amount" spot because 7x represents the total cost of making the 7 bags.
7x ≤ 42
Now we can solve. Since a constant is multiplied by a variable, you will use division. You need to find a number you can divide both sides by that will leave x by itself. If you divide by the constant, the multiplication and division by the same number will cancel out to 1, leaving 1x, or x, alone on the left side. Divide both sides by 7:
x ≤ 6 is your final answer.
Now turn this inequality back into words. Remember that x represents the cost for each bag, and ≤ represents "less than or equal to." So, what this inequality is saying is that the cost to make each bag has to be less than or equal to $6.
Determining if a Given Value Makes an Inequality True
The last type of problem we are going to talk about give you an inequality with a variable in it, and a number that could replace that variable. It's your job to determine if that number makes an inequality true. All you have to do is solve the inequality, and then replace the variable with the new number
Consider this problem:
-5x ≤ 45. Would an x value of 4 make this inequality true?
First, let's solve. A constant (-5) is multiplied by a variable (x), so we will need division to undo that multiplication. Since we are multiplying by -5, if we divide both sides by -5 the multiplication and division will cancel out:
x ≥ -9
Now replace the variable (x) with the new value (4).
4 ≥ -9
This is a true statement, so the answer is yes, an x value of 4 would make the inequality true.
It's good to solve first and then replace the variable with the new value because you can use the same simplified inequality for any other numbers you want to try, but you can also just plug the new value straight into the original equation. Consider this problem:
x + 7 > 10. Would an x value of 3 make this inequality true?
If you plug 3 straight into x, the inequality will end up looking like this:
3 + 7 > 10
10 > 10
10 is equal to 10, but it is not greater than 10, so an x value of 3 would not make this inequality true.
If this way of approaching the problem is easier for you to understand, then definitely do it instead.
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