1. What connections can we make between the solution of an inequality and the graph of the inequality?
2. Provide a scenario for each of the inequality symbols. (For example- Sally lives less than 5 minutes away, m < 5)
3. When does the inequality symbol flip when solving an inequality?
4. What similarities and differences do equations and inequalities have? Give two examples of each.
Submit your answers below for the Unit 4 Essential Questions Quiz.
Some common mistakes that are made with inequalities are as follows:
1) Forgetting to switch the inequality symbol when multiplying or dividing by a negative number.
This was covered in a previous journal entry.
Ex. 5 - x > 3 will solve to -x > -2 so we multiply or divide by a -1 on both sides to get x to become positive. This becomes x > 2. We have to flip the sign because we multiplied or divided by the negative so the final inequality will be x < 2. To make sure that it is true, we can plug in numbers on either side of 2 to show why we have to flip the sign.
With Flipping the Sign: Pick a number smaller than 2. Plug it into the original inequality 5 -x > 3. Is the inequality true when solved?
Without Flipping the Sign: Pick a number bigger than 2. Plug it into the original inequality 5 -x > 3. Is the inequality true when solved?
You should see that without flipping the sign, the inequality is no longer true so we must flip the sign in order to make it true.
2) Graphing the inequality wrong by shading in the wrong direction when the variable is on the right side of the inequality
Use the symmetric property to rewrite the inequality how you're used to seeing it to avoid this mistake.
If you're given a problem like -3 < x + 1 and you solve it to get -4 < x, this means that -4 is smaller than x so x must be bigger than -4. Using the symmetric property, we can see that it is true by writing x > -4. The inequality symbol is still open toward the x as it was in the original problem. This means we should have an open circle on -4 and shade to the right to show that all numbers equal to or greater than -4 are possible solutions to the inequality.
3) Solving the inequality with a > or < sign and identifying the final answer as a possible answer. If it doesn't say ≤ or ≥ then the final answer can't be a solution.
Ex. 5 + x > 3 will become x > -2 when you solve it, but -2 is not a solution because the answers must be greater than -2, so -1.9999 is an answer, but not -2.
4) Shading in the wrong direction when you have an inequality with a negative number
This mistake is not very common, but sometimes you may get confused when you have
x < -3 and you will write the numbers incorrectly on the number line or you may shade to the right instead of the left because of a misunderstanding of which numbers are smaller when dealing with negatives.
Solving inequalities is very similar to solving equations.
Solving Inequalities Practice Problem: x + 5 ≥ -3
Use the inverse operation to isolate the variable by - 5 on both sides
x ≥ -8
The inequality has been solved. If asked for possible solutions, you must say that the answer could be any number equal to or greater than -8. Pay attention to the inequality symbols as they tell you exactly which numbers can be solutions.
The main difference to note between solving inequalities and solving equations is that when dividing or multiplying by a negative number (when you have a negative coefficient for your variable), you must flip the inequality symbol to its opposite to keep the inequality true. See the videos below for examples.
Practice Problems in Class:
(1) The product of nine and x is greater than six more than the product of three and x.
(2) Mrs. Scott decided that she would spend no more than $120 to buy a jacket and a skirt. If the price of the jacket was $20 more than 3 times the price of the skirt. Find the highest possible price of the skirt?
(3) Stephanie weighs 3 times as much as Rachel. Both weights are whole numbers and the sum of their weights is less than 160 pounds. Find the greatest possible weight for each girl.
(4) Six more than two times a certain number is less than the number increased by twenty. Find the numbers that satisfy this condition.
(5) Two consecutive even integers are such that their sum is greater than 98 decreased by twice the larger. Find the smallest possible values for the integers.
(6) Mrs. Smith wrote "Eight less than three times a number is greater than fifteen" on the board. If x represents the number, write an inequality that is a correct translation of this statement.
The documents below are handouts for practice problems (not all problems to be completed) reviewed in class as classwork or homework. Notes are provided in class to add to the handouts or the journal entries.
An inequality shows a relationship between two or more expressions (or terms) that is not equal. There are numerical inequalities and algebraic inequalities. Numerical inequalities use only numbers while algebraic inequalities use variables to compare. The solution set shows the answers that make the inequality true when it is solved and/or graphed. Inequalities can have one of five different signs. See the image below for how to graph each of the signs, how to write the inequality (both ways using the symmetric property), and key words that represent that particular sign
When you have a variable in an inequality, the variable represents all the numbers that are possible answers, called the solution set. These answers are being compared to the other number in the inequality. For example in x < 1, the solution set is any number less than 1 but not 1 or any number higher. Therefore if we needed to find possible solutions for x < 1, we would say any number lower than 1 is a solution. This is important to remember for solving inequalities in the next journal entry.
The difference between "is less than", "less than" and "less".
Notice that in inequality word problems, the word "is" (or some variation of it like has or was) is present. However, sometimes we can confuse it with the meaning for "less than" or "less". Remember that is means = sign, but if something IS less than then it is not equal.
If, instead you see, 5 less than 10, that means to subtract because the word IS isn't there. So the expression would be 10-5 for 5 less than 10. The word than makes us reverse the order of the term being subtracted.
Lastly, 5 less 10 still means to subtract, but the expression would be 5 - 10.
The worksheet below was used during class to provide practice problems on writing and graphing inequalities.