# Solving Inequalities: Tips & Tricks for Beginners

• ezsmith
In summary, when dealing with inequalities, the direction of the inequality changes when you multiply or divide by a negative number. This includes taking the reciprocal of each side. It is helpful to treat the inequality like an equal sign and always check your answer by plugging it back in.
ezsmith
Hi, I used to do all this type of inequalities but I have not practice this for almost a year and I have totally forgotten and didn't know why did the sign change.

For example: If it's 3x ≤ 9

Is this right? I have actually forgotten in what circumstances we are suppose to change the inequalities. Input and explanation will be much appreciation. Thanks!

Last edited by a moderator:

It's actually the other way around, i.e., x <= 3.
You can always test your answer. For example if x = 1, then 3x <= 9. But x is not greater than 3. Thus your answer is wrong.

You only change the inequality when the coefficient is negative ( -x > 0 <=> x < 0).

Rikardus said:
It's actually the other way around, i.e., x <= 3.
You can always test your answer. For example if x = 1, then 3x <= 9. But x is not greater than 3. Thus your answer is wrong.

You only change the inequality when the coefficient is negative ( -x > 0 <=> x < 0).
More precisely, the direction of the inequality changes when you multiply both sides of the inequality by a negative number. This includes division, as well, since division by a number is the same as multiplying by the reciprocal of that number.

The direction of the inequality changes if you take the reciprocal of each side. For example, 2 < 3, but 1/2 > 1/3.

All of these replies got it right. What helped me when I was learning about inequalities, was to treat it like an equal sign. Then the only additional thing to remember when solving them is to change the direction of the inequality any time you multiply or divide the equation by a negative number.

Also, as was pointed out above. you can always check to make sure the answer makes sense by plugging it back in.

good luck!

Hello, thank you for reaching out. Yes, your answer is correct. When solving inequalities, there are a few key things to keep in mind. First, remember that the direction of the inequality sign (≤ or ≥) does not change when you multiply or divide by a positive number. So in the example you provided, 3x ≤ 9, you can divide both sides by 3 to get x ≤ 3. However, if you multiply or divide by a negative number, the direction of the inequality sign does change. In this case, you would need to flip the sign. For example, if you had -3x ≥ 9, you would divide both sides by -3 and change the sign to get x ≤ -3. It's also important to remember to keep the inequality sign pointing towards the smaller number. In your example, 3x ≤ 9, the smaller number is 3, so the sign remains pointing towards that number. I hope this helps refresh your memory and provides some clarity on solving inequalities. Keep practicing and you'll soon have it down again. Best of luck!

## 1. How do you solve inequalities?

Solving inequalities involves finding the values that make the statement true. This can be done by using algebraic manipulation or graphing the inequalities on a number line.

## 2. What are the different types of inequalities?

The three main types of inequalities are linear inequalities, quadratic inequalities, and rational inequalities. Each type involves solving for a variable in an inequality statement.

## 3. What are some tips for solving inequalities?

Some tips for solving inequalities include: simplifying the expressions, keeping track of the direction of the inequality sign, using inverse operations to isolate the variable, and checking for extraneous solutions.

## 4. How do you graph inequalities?

To graph inequalities, first solve for the variable, then plot the solutions on a number line. Use an open circle for less than or greater than inequalities, and a closed circle for less than or equal to or greater than or equal to inequalities. Shade the region to the right or left of the number line, depending on the direction of the inequality sign.

## 5. What are some common mistakes when solving inequalities?

Some common mistakes when solving inequalities include forgetting to change the direction of the inequality sign when multiplying or dividing by a negative number, forgetting to check for extraneous solutions, and not fully simplifying the expression before solving for the variable.

Replies
11
Views
2K
Replies
4
Views
2K
Replies
15
Views
1K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
4
Views
1K
Replies
2
Views
2K
Replies
1
Views
816
Replies
2
Views
2K
Replies
5
Views
905