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.