Absolute Value Inequalities - Algebra

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, can never happen. There is no solution.

Split the inequality into two possible cases as follows, based on the absolute values.

First case:

Second case:

Let's find the inequality of the first case.

Multiply both sides by x + 6.

Subtract x from both sides, then subtract 3 from both sides.

Divide both sides by 3.

Let's find the inequality of the second case.

Multiply both sides by x + 6.

Simplify.

Add x to both sides, then subtract 3 from both sides.

Divide both sides by 5.

So the range of x-values is and .

Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.

First, subtract 5 from both sides to get the absolute value expression alone.

Split this into two linear equations:

or

The solution set is

The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.

or

Solve each inequality separately by adding to all sides.

or

This can be simplified to the format .

Remove the absolute value by setting the term equal to either or . Remember to flip the inequality for the negative term!

Solve each scenario independently by subtracting from both sides.

The absolute value of any number is nonnegative, so must always be greater than . Therefore, any value of makes this a true statement.

The absolute value of a number must always be nonnegative, so can never be less than . This means the inequality has no solution.

First, we can simplify this inequality by subtracting 7 from both sides. This gives us

Next, however, we need to make two separate inequalities due to the presence of an absolute value expression. What this inequality actually means is that

and

(Be careful with the inequality signs here! The second sign must be switched to allow for the effect of absolute value on negative numbers. In other words, the inequality must be greater than because, after the absolute value is applied, it will be less than 7.) When we solve the two inequalities, we get two solutions:

and

For the original statement to be true, both of these inequalities must be fulfilled. We're left with a final answer of

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.

--

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

Check with

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:

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, can never happen. There is no solution.