Solving Inequalities - Algebra II

For a compound inequality, we solve each inequality individually. Thus, for the first inequality, , we obtain the solution and for the second inequality, , we obtain the solution . In interval notation, the solutions are and , respectively. Because our compound inequality has the word "or", this means we union the two solutions to obatin .

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 .

Add 4 to both sides.

Divide both sides by –7. When dividing by a negative value, we must also change the direction of the inequality sign.

The first step is to distribute (multiply) through the parentheses:

Then subtract from both sides of the inequality:

Next, subtract the 12:

Finally, divide by two:

or, in interval notation,

or, in interval notation,

Inequalities can be treated like any other equation except when multiplying and dividing by negative numbers. When multiplying or dividing by negative numbers, we just flip the sign of the inequality so that becomes , and vice versa.

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain equation of the boundary:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - for ease, we will choose :

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0 is less than 7 so the correct symbol is .

The correct choice is .

Let be Sam's age, and let be his brother's age.

In the problem, we are told that the sum of their ages is at least 18. Represent this with an inequality:

Sam's age is three years more than twice his brothers age. Write this mathematically as:

Plug in for the value in the inequality and solve for :

The age of Sam's brother is less than or equal to years.

Start by subtracting 1 and divinding by 4 on both sides of the equality

Written in interval notation:

In order to solve this inequality, we must first consolidate all of our values on one side.

The first thing we need to do is move the to the other side:

This results in:

Next, we need to move the from the right side over to the left side:

This gives us

Dividing each side by gives us our solution:

The two equations should be solved separately to get,

and

.

This can be checked by plugging in values between and and seeing if they satisfy both equations.

Most of solving inequalities is straightforward algebra and we can manipulate them in the same way as equations in most cases.

However, we must remember that when multiplying or dividing by negative numbers in inequalities, we have to switch the direction of the inequality. So we do the final division step and get the answer:

Remember: Use inverse operations to undo the operations in the inequality (for example use a subtraction to undo an addition) until you are left with the variable. Make sure to do the same operations to both sides of the inequality.

Important Note: When multiplying or dividing by a negative number, always flip the sign of an inequality.

Solution:

Expand all factors

Simplify

Add 23

Subtract 22m

Divide by -6 (We flip the sign of the inequality)

Simplify

First, add 2 to both sides of the inequality:

and simplify: .

Then, multiply each side by 3:

and simplify: .

When solving an inequality, first isolate the variable:

(subtract 5 from both sides)

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(divide both sides by -2)

(remember when dividing both sides by a negative, you must flip the inequality sign because the sign on both sides changed)

is the answer!

Important note:

The negative two cancels on the right side and the on the left side. Since both sides went from negative to positive values the inequality sign flips.

Isolate all the terms with x on one side and all other terms on the other side. Our first step is to subtract four from each side.

We the get

.

We now need to divide both sides by -5.

However, whenever you multiply or divide by a negative number, you flip the direction of the inequality.

To make this problem easier to solve, we can add 2 to both sides so that we can factor the left side of the expression.

The breakpoints to examine are at

These two breakpoints create 3 total regions that we need to examine:

, , and . Which ever region satisfies the expression above will be a solution to the inequality.

A value of -3 gives us: .

is greater than 0, so it satisfies the inequality.

A value of -1.5 for the second region does not satisfy the inequality.

A value of 0 for the third region does satify the inequality, so the first and third regions give us our answer.

.

To solve the inequality we want to isolate the x variable on one side and all other constants on the other side.

The first step is two add six to both sides.

Next, divide by negative four and remember when dividing by a negative you must flip the inequality sign.