How to Solve Absolute Value Inequalities
Evaluate the form of the absolute value inequality., Transform an absolute value inequality into normal inequalities., Ignore the inequality sign while you solve for x for the first equation., Solve as you would normally for x., Write the solution...
Step-by-Step Guide
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Step 1: Evaluate the form of the absolute value inequality.
As mentioned above, the absolute value of x, denoted by βπ₯β, is defined as:
Absolute value inequalities usually take one of the following forms: βπ₯β< or βπ₯β>πΒ ; βπ₯Β±πβ<π or βπ₯Β±πβ>πΒ ; βππ₯2 +ππ₯β<π In this article, the focus will be on inequalities of the form |f(x)|<a or |f(x)|>a , where π(π₯) is any function, and a is a constant. -
Step 2: Transform an absolute value inequality into normal inequalities.
Remember that an absolute value of x can either be positive x or negative x.
The absolute value inequality βπ₯β< 3 can also be transformed into two inequalities:
-x < 3 or x <
3.
For example,βxβ3β>5 can be transformed into β (π₯β3)>5 or π₯β3>5. β3π₯+2β <5 can be transformed into β (3π₯+2)<5 or 3π₯+2<5.
The term βorβ means that either of the two will satisfy the given absolute value problem. , If it helps, temporarily replace the inequality sign with an equal sign until the end. , Remember that if you divide by a negative number to isolate x on one side of the inequality sign, you will also have to switch the inequality sign.
For example, if you divide both sides by
-1,
-x>5 will become x<-5. , From the values above, you need to write the range of values that can be substituted in for x.
This range of values is often referred to as a solution set.
Since you would have to solve two inequalities from the absolute value inequality, you will then have two solutions.
In the example used above, the solution can be written in two ways:
-7/3<x<1 (-7/3,1) , Pick a number within your solution set, and plug it in for x.
If it works, great! If not, go back and check your arithmetic. -
Step 3: Ignore the inequality sign while you solve for x for the first equation.
-
Step 4: Solve as you would normally for x.
-
Step 5: Write the solution set.
-
Step 6: Check your work.
Detailed Guide
As mentioned above, the absolute value of x, denoted by βπ₯β, is defined as:
Absolute value inequalities usually take one of the following forms: βπ₯β< or βπ₯β>πΒ ; βπ₯Β±πβ<π or βπ₯Β±πβ>πΒ ; βππ₯2 +ππ₯β<π In this article, the focus will be on inequalities of the form |f(x)|<a or |f(x)|>a , where π(π₯) is any function, and a is a constant.
Remember that an absolute value of x can either be positive x or negative x.
The absolute value inequality βπ₯β< 3 can also be transformed into two inequalities:
-x < 3 or x <
3.
For example,βxβ3β>5 can be transformed into β (π₯β3)>5 or π₯β3>5. β3π₯+2β <5 can be transformed into β (3π₯+2)<5 or 3π₯+2<5.
The term βorβ means that either of the two will satisfy the given absolute value problem. , If it helps, temporarily replace the inequality sign with an equal sign until the end. , Remember that if you divide by a negative number to isolate x on one side of the inequality sign, you will also have to switch the inequality sign.
For example, if you divide both sides by
-1,
-x>5 will become x<-5. , From the values above, you need to write the range of values that can be substituted in for x.
This range of values is often referred to as a solution set.
Since you would have to solve two inequalities from the absolute value inequality, you will then have two solutions.
In the example used above, the solution can be written in two ways:
-7/3<x<1 (-7/3,1) , Pick a number within your solution set, and plug it in for x.
If it works, great! If not, go back and check your arithmetic.
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