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Solving Inequalities: How to Use Greater Than and Less Than Signs


Before learning how solving inequalities works, we need to know what the direction of the inequality symbols mean in the following inequality signs:







Now let's show you how solving inequalities is different from solving equations.


What Are Inequalities?


Solving inequalities is very similar to solving regular math equations. The only difference is that they use inequality symbols instead of equal signs:


Math Equations:






Math Inequalities:






There are two sides of an inequality but in order to solve an inequality, we want to get the variable of x on the left side. This will require some subtracting, adding, multiplying, and dividing.



In the example above, the inequality x is on the right-hand side. We want to solve for the x and switch its sides of the inequality.


How Does Solving Inequalities Work?


You must use this multi-step process to solve the practice problem below:



1. Add or Subtract Both Sides


We'll want to add 2 to both sides. This will cancel out the value of negative 2. If the 2 was a positive number. However, we would do the inverse and subtract this value from both sides.






2. Divide by the Coefficient


In order to solve for x, we need to divide by -4.



But since this value is a negative number we'll have to...


3. Change the Direction of the Inequality Sign


When you divide or multiply by a negative number, you must change the direction of the inequality sign. This is because of this rule of inequalities:



As you can see, x is originally less than y. But, when it’s multiplied by a negative number, x becomes greater than y. That’s because smaller negative numbers are closer to 0, making them worth more. That’s why, when multiplied by a negative number, the larger of the two values will be less than the smaller of the two values.


So since we've divided by negative 4, the rules of inequalities state we must flip the inequality symbol:




You also change the direction of the inequality symbol when you reverse the direction of the equation. This is basically flipping it so that the variable 4x and constant of -2 is on the other side of the equation:



Solving Inequalities


Solving inequalities is like solving regular equations except you use inequality signs:


.

You have to change the direction of these signs when you divide or multiply by a negative sign or flip the side of the equation the variable is on.


All in all, learning how to solve inequalities can help improve your understanding of how greater than or less than signs work.


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