What is the purpose of negative numbers

Solving Inequalities: 3 Ways to Solve Inequalities

First of all: Solving inequalities is no easier or more complicated than solving equations. I can reassure you about that. The work steps are largely the same.

First of all, what is an inequality? An inequality looks much the same as an equation with a difference. The is-equal sign is replaced by an inequality sign. This is a small triangle with the tip pointing either to the left or to the right. You can see this in the following examples:

Equation: 2x = 3 - x

Inequality: 2x <3 - x

The goal is the same in both cases. Solving inequalities, like solving equations, means that you want to solve for x. To do this, you use the same steps for solving inequalities as for equations. The result also looks the same. The difference is again just the inequality sign:

Equation: x = 1

Inequality: x <1

I want to make you understand that you don't have to do much differently when solving inequalities than when solving equations. First take a look at the following video. If you then still have difficulties with solving inequalities, I will explain in more detail what you have to watch out for when solving inequalities and where the most important sources of error in classwork are hidden.

Solving inequalities: explanatory video

This video gives you detailed explanations about solving inequalities.

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Solving inequalities: what tools do I need?

Let's reiterate that inequalities are very similar to equations. How to correctly solve equations can be repeated and deepened on the LEARNZEPT.de website using detailed instructional videos and practiced through real, interactive classwork.

The key difference between equations and inequalities concerns the inequality sign. When solving inequalities, you don't get a single value as a solution, but a set of numbers that meet the conditions (> “greater than” / <“less than”) of the solution.

You therefore have to specify a so-called solution set as the solution of an inequality. Let's look at an example of solving an inequality:

x> 2

So the solution for x is “all values ​​that are greater than 2”. You can write down the solution set for solving inequalities in two ways:

  1. Set notation: IL = {x I x> 2}
  2. Interval notation: IL =] 2; ∞ [

You urgently need to memorize one of the most important arithmetic rules when solving inequalities, which is not necessary when solving equations. If you divide an inequality by a negative number or take it with a negative number, then the inequality sign is reversed in the same step. Would you like an example?

5x> 10 I: 5 (divide by a positive number!) X> 2 (sign remains the same!)

-5x> -10 I: -5 (divide by a negative number!) X <2 (sign reverses!)

The problem with the sign and the reversal of the sign when you multiply with a negative number, you know in simpler equations by solving parentheses. You can find more details on this on the LEARNZEPT.de page.

This change in the inequality sign is called inversion. Looks complicated, but it's not that tragic because you only have to reverse the sign when solving inequalities if you multiply with a negative number or divide by a negative number.

In the following, I'll point out the most common mistakes students make when solving inequalities in my experience and give you tips on how to best avoid these mistakes.

 

Solving inequalities: the mistakes are in the details!

  1. Inversion ignored

Unfortunately, when solving inequalities, students often forget that the inequality sign changes if you divide by negative numbers or multiply them.

My tip: Just always remember that negative numbers are bad when solving inequalities and then look carefully to see whether you have taken the inversion into account when taking and dividing with negative numbers!

  1. Everything calculated correctly, but the amount of solutions written down incorrectly

When correcting class work on the topic of solving inequalities, I have often wondered how unnecessarily students give away points because they did not write down the solution set correctly. The problem has been calculated correctly and the result is also correct, as in the example: x> 2

Wrong notation: IL = x> 2

Correct notation: IL = {x I x> 2} or: IL =] 2; ∞ [

 

My tip: When writing down the solution set, choose one of the two notations (set notation or interval notation) and then use it reliably. You can find more details about the different spellings of quantities on the LEARNZEPT.de page.

  1. Difference between and ≥

Another mistake in solving inequalities that often happens to students is that when writing down the solution set, they do not pay attention to whether the inequality sign has a line below it. This means that the limit must be included in the solution set, since the value can not only be larger or smaller but can also be the same.

My tip: The easiest way to avoid this careless mistake is to double-check your solution after solving the inequality and before writing down the solution set.

  1. Basic amount not taken into account

The last mistake I would like to point out when solving inequalities is that students often forget that the given basic set of the problem can limit the set of solutions. For example, if you have specified basic sets such as “natural numbers” or “whole numbers”, you may not include decimal numbers or fractions in the solution set. If your basic set only allows positive numbers, then you must not include negative numbers in the solution set.

My tip: As stupid as it sounds, only one thing helps here: Take a close look at the basic set and consider whether it might limit your solution set.

 

Solving inequalities: Lastly, 3 tips

  1. Solving inequalities works like solving equations! (Caution inversion!)
  1. When you have solved the inequality, write down the solution set! (Decide on a spelling and stick to it!)
  1. Pay attention to whether the basic amount limits your amount of solution!

 

Solve inequalities: You can get help here

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