Resolver desigualdades de valor absoluto

Resolver desigualdades de valor absoluto

In this lesson, we are going to learn how to solve absolute value inequalities using the standard approach usually taught in an algebra class. That is, learn the rules and apply them correctly. There are four cases involved when solving absolute value inequalities.

PRECAUCIÓN: In all cases, the assumption is that the value of “a” is positive, that is, a > 0.

Four (4) Cases to Consider When Solving Absolute Value Inequalities

CASO 1:



Resolver desigualdades de valor absoluto Resolver desigualdades de valor absoluto

CASO 2:


Resolver desigualdades de valor absoluto Resolver desigualdades de valor absoluto

CASO 3:



Resolver desigualdades de valor absoluto

The absolute value of any number is either zero (0) or positive which can never be less than or equal to a negative number.

The answer to this case is always sin solución.

CASO 4:

Resolver desigualdades de valor absoluto

The absolute value of any number is either zero (0) or positive. It makes sense that it must always be greater than any negative number.


The answer to this case is always todos los números reales.

Examples of How to Solve Absolute Value Inequalities

ejemplo 1: Solve the absolute value inequality .

Resolver desigualdades de valor absoluto

If you’re not familiar yet with the different cases, I suggest that you keep a copy of the list of cases above as a reference. This will definitely help you solve the problems easily.


The problem suggests that there exists a value of “x” that can make the statement true. Well, the absolute value of something is always zero or positive which is never less than a negative number. This statement must be false, therefore, there is sin solución. Esto es un ejemplo de caso 3.

Pick some test values to verify:

  • If x is positive, say, x = 5;
Resolver desigualdades de valor absoluto
  • If x is zero;
Resolver desigualdades de valor absoluto
  • If x is negative, say, x = -5;
Resolver desigualdades de valor absoluto

ejemplo 2: Solve the absolute value inequality .


Resolver desigualdades de valor absoluto

If you think about it, any values of “x” can make the statement true. Test some numbers including zero, and any negative or positive number. What do you get?

Remember, the absolute value expression will yield a zero or positive answer which is always greater than a negative number. Therefore, the answer is todos los números reales. Es caso 4.

ejemplo 3: Solve the absolute value inequality .

Resolver desigualdades de valor absoluto

This is a “less than” absolute value inequality which is an example of caso 1. Get rid of the absolute value symbol by applying the rule. Then solve the linear inequality that arises.

Resolver desigualdades de valor absoluto

The goal is to isolate the variable “x” in the middle. To do that, we subtract the left, middle and right parts of the inequality by 6.

Resolver desigualdades de valor absoluto

The answer in the form of the inequality symbol states that the solutions are all the values of x between -8 and -4 but not including -8 and -4 themselves.

We can also write the answer in interval notation using a parenthesis to denote that -8 and -4 are not part of the solutions.

Resolver desigualdades de valor absoluto

Or, write the answer on a number line where we use open circles to exclude -8 and -4 from the solution.

Resolver desigualdades de valor absoluto

ejemplo 4: Solve the absolute value inequality .

Resolver desigualdades de valor absoluto

This is a “less than or equal to” absolute value inequality which still falls under caso 1. Clear out the absolute value symbol using the rule and solve the linear inequality.

Resolver desigualdades de valor absoluto

Isolate the variable “x” in the middle by adding all sides by 6 and then dividing by 3 (coefficient of x).

Resolver desigualdades de valor absoluto

The inequality symbol suggests that the solution are all values of x between -3 and 7, and also including the endpoints -3 and 7. We include the endpoints because we are using the symbol “≤“.

To write the answer in interval notation, we will utilize the square brackets instead of the regular parenthesis to denote that -3 and 7 are part of the solution.

Resolver desigualdades de valor absoluto

And finally, we will use closed or shaded circles to show that -3 and 7 are included.

Resolver desigualdades de valor absoluto

ejemplo 5: Solve the absolute value inequality .

Resolver desigualdades de valor absoluto

This is an example of a “greater than” absolute value inequality which is an example of caso 2. Let’s eliminate the absolute value expression using the rule below.

Resolver desigualdades de valor absoluto

As you can see, we are solving two separate linear inequalities.

Resolver desigualdades de valor absoluto

or

Resolver desigualdades de valor absoluto

In interval notation, the word “or” is replaced by the symbol “ cup ” to mean “unión“. The union of sets means that we are putting together the non-overlapping elements of two or more sets of solutions.

Resolver desigualdades de valor absoluto

The answer in interval notation makes more sense if you see how it looks on the number line. In case 2, the arrows will always be in opposite directions. The open circles imply that -3 and 7 are not included in the solutions which are the consequence of the symbol “>“.

Resolver desigualdades de valor absoluto

ejemplo 6: Solve the absolute value inequality .

Resolver desigualdades de valor absoluto

Break this up into two linear inequalities, and then solve each separately. Here’s the rule for caso 2.

Resolver desigualdades de valor absoluto

Esta es la solución.

Resolver desigualdades de valor absoluto

or

Resolver desigualdades de valor absoluto

For the interval notation, we use the square brackets to include -2 and 3 in the solution.

Resolver desigualdades de valor absoluto

The shaded or closed circles signifies that -2 and 3 are part of the solution. In case 2, the arrows will always point to opposite directions.

Resolver desigualdades de valor absoluto

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