Inequalities involving  generate two separate inequalities and can't be combined into a single inequality.

First isolate the absolute value expression on one side of the inequality

Subtract eight from each side.

From here separate the expression into two expressions for which we will need to solve.

Add six to get side.

To solve an inequality with absolute value you have to consider the two equations it creates.

becomes  and

Solve for both inequalities by following the balancing rules. Be careful of division or multiplication of a negative number; if that happens, flip the inequality sign.

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We first need to eliminate the absolute value sign by making two inequalities: 

 and . Remember that when the number becomes negative we must flip the inequality symbol.

From there, it is a simple one-step inequality. We divide both sides by  to get:

.

Because x lies between  and , we can combine these into one inequality:

Split up the absolute value and rewrite the inequalities in two formats:

Solve the first inequality.  Subtract three on both sides.

Divide by two on both sides.

The first solution is:  

Evaluate the second inequality.  

Divide negative one on both sides.   Dividing a negative term will switch the sign.

Subtract three on both sides.

Divide by two on both sides.

The second solution is:  

In interval notation, the answer is:  

Solve by first dividing three on both sides.  We will need to isolate the absolute value sign before splitting the absolute value into its positive and negative components.

Split the absolute value into two equations.

Solve the first equation.  Subtract three on both sides and simplify.

Divide by three on both sides.

This is the first solution.  Solve the second equation.  Divide by a negative one on both sides.  This will switch the inequality sign.

Subtract three from both sides and simplify.

Divide by three on both sides.

The answer is between negative five and three but does not include both numbers:  

Divide by negative three on both sides.  Dividing by a negative number will switch the sign.

Simplify the fractions and switch the sign.

We will notice that this has no solution because absolute value converts all values to a positive number.  There are no such values inside an absolute value that will be less than negative two.

The answer is:  

Split the absolute value inequality into its positive and negative components.

Solve the first inequality.  Subtract three on both sides.

Divide by three on both sides.

This is one of the solutions.

Simplify the second inequality.  Divide by negative one on both sides and switch the sign.

Subtract three on both sides to isolate the x term.

Divide by three on both sides.

The second solution set is:  

The solution is only valid from:  

Split the absolute value into its positive and negative inequalities. Be sure to encapsulate the left quantity with a negative sign for the second inequality.

First inequality: 

Second inequality: 

Solve the first inequality. Add  on both sides.

Subtract seven from both sides.

Divide by five on both sides.

Simplify both sides.

The first solution is:

Solve the second inequality. First divide by a negative one on both sides and switch the inequality sign.

Simplify both sides and change the sign.

Subtract  from both sides.

Subtract seven from both sides.

The answer in interval notion is: 

Break the inequality into two parts:

For (2), because we multiplied one side by -1 we must flip the inequality sign. In general, when you multiply one side of an inequality by a negative number, the sign must be flipped. 

AND

Final Solution  

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