Inequalities are math expressions in which both sides are not equal to each other. In an equation, both sides are equal to each other. For example, "2q + 2 = 4" is an equation, while "2q + 2 < 4" is an inequality because the expressions are not equal. Another difference is that equations usually have at least one answer, while inequalities have infinite.
There are different symbols that signify different inequalities. The greater than symbol (>) signifies that the first equation is larger than the other, and the less than symbol (<) signifies that the first expression is smaller than the other. The greater than or equal to symbol (≥) signifies that the first expression is either equal to or greater than the other, and the less than or equal to symbol (≤) signifies that the first expression is either equal to or smaller than the other.
To solve an inequality you have to use algebra. An example of this can be "n > 6". In this inequality, "n" is greater than 6, so "n" can't be six. Based on this information, "n" can be anything greater than 6 (such as 400). We know this because of the sign. Another example of this can be "5p + 25 ≤ 125". First, you will subtract 25 from 125 to get 100. Next, you will divide both sides by 5 to isolate the variable. Don't forget that the sign stays the same. After this, you will see that "p ≤ 20". You can always work backwards to check your work!
There is only one exception to the instructions above. The sign switches sides when multiplying by a negative number. An example of this "-5p + 2 ≤ 7". First, you will subtract both sides by 2. Then, you will be left with "-5p ≤ 5". Next, you will divide both sides by -5 to isolate the variable. Because you have divided both sides by a negative number, the sign will switch sides. Lastly, you will be left with "p ≥ -1".
Certain types of inequalities are unsolvable. One of these unsolvable inequalities is "y > y + 1". No matter what the value of "y" is, the inequality is unsolvable. To make it easier and clearer, you can simplify the inequality. If you do so, you will get "0 > 1", which is false. Because of this, this inequality is deemed as false, or unsolvable.
As you already know, inequalities have more than one answer. You can't write down every single number that can be substituted for the variable, but you can graph the values to show the same. To start graphing, you will need a number line. Here's how to graph inequalities. Make sure to simplify each inequality before graphing! First, you need to identify the sign. The ">" and "<" signs are shown with an empty dot, and the "≥" and "≤" signs are shown with a colored dot. The dot goes on top of the value provided in the inequality (such as in "n > 6", an empty dot would go over 6). Next, to show the type of inequality (greater than, less than, etc.), you will add a line with an arrowhead at the end of it depending on the sign. For the ">" and "≥" signs, the line goes to the right. For the "<" and "≤", the line goes to the left. For example, in "n > 6", there would be a line going to the right from the empty dot. Remember, you can only graph properly when your isolated variables are on the left side of the inequality.
An example of this is "-5n ≤ 25", which can be simplified to "n ≥ -5". For this example, since the sign is a "≥", there will be a shaded dot on -5. Since "n" is greater than or equal to -5, the arrow will be pointing towards the left. Another example is "2j - 10 < -4". If you simplify this, you will get "j < 3". Since "j" is less than 3, there will be an empty dot over 3 pointing to the left. The answers to both of these inequalities can be any number in the number line which has the arrow on top of it because the arrow signifies the possible answers.
Word problems are a part of everything in math. This includes inequalities! Inequality word problems are interesting and fun to solve!
Here's an example of one: "Bryce is thinking to work at a hotel. Hotel A gives him $16 an hour, but it takes $14 of gas to get there and back. Hotel B is right next to his house but only gives him $9 an hour. What is the least number of hours he will need to work to get more money from Hotel A?"
To start solving this, we must identify our equation. Our equation will be "16s -14 > 9s". The variable "s" will represent the number of days Bryce stayed. Next, we just simplify the equation. After simplifying you'll get "s > 2", which means Bryce would have to work at least 3 hours to make Hotel A's price larger than Hotel B.
Inequalities are needed everywhere! You've probably not noticed them. You're also probably living under a rock.
Here's an example. When you're driving on a road, there's a speed limit. You always have to drive below the speed limit (or five over). If the speed limit is 60 kilometres an hour, the inequality would be "s < 60". The variable "s" represents speed.
Inequalities are really important to your everyday life! They're needed from speed limits to curfews to even ice cream! Next time you decide to get out from under your mental rock, try to find inequalities around you!