# SAT II Math II : Solving Inequalities

## Example Questions

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### Example Question #1 : Solving Inequalities

Give the solution set of the inequality:

Explanation:

First, find the zeroes of the numerator and the denominator. This will give the boundary points of the intervals to be tested.

Either

or

Since the numerator may be equal to 0,  is included as a solution; , since the denominator may not be equal to 0,  and  are excluded as solutions.

Now, test each of four intervals for inclusion in the solution set by substituting one test value from each:

Let's test :

This is true, so  is included in the solution set.

Let's test :

This is false, so  is excluded from the solution set.

Let's test :

This is true, so  is included in the solution set.

Let's test :

This is false, so  is excluded from the solution set.

The solution set is therefore .

### Example Question #1 : Solving Inequalities

Give the solution set of the inequality:

Explanation:

Put the inequality in standard form, then

Find the zeroes of the polynomial. This will give the boundary points of the intervals to be tested.

or .

Since the inequality is exclusive (), these boundary points are not included.

Now, test each of three intervals for inclusion in the solution set by substituting one test value from each:

Let's test :

This is false, so  is excluded from the solution set.

Let's test :

This is false, so  is excluded from the solution set.

Let's test :

This is true, so  is included in the solution set.

The solution set is the interval .

### Example Question #1 : Solving Inequalities

Solve the inequality:

Explanation:

Subtract  on both sides.

Simplify both sides.

Divide by negative five on both sides.  This requires switching the sign.

### Example Question #1 : Solving Inequalities

Solve:

Explanation:

The first thing we can do is clean up the right side of the equation by distributing the , and combining terms:

Now we can combine further. At some point, we'll have to divide by a negative number, which will change the direction of the inequality.

### Example Question #1 : Solving Inequalities

Solve: .

Explanation:

First, we distribute the  and then collect terms:

Now we solve for x, taking care to change the direction of the inequality if we divide by a negative number:

### Example Question #1 : Solving Inequalities

Solve:

Explanation:

In order to solve this inequality, we need to apply each mathematical operation to all three sides of the equation.  Let's start by subtracting  from all the sides:

Now we divide each side by .  Remember, because the  isn't negative, we don't have to flip the sign:

### Example Question #1 : Solving Inequalities

Solve the inequality:

Explanation:

Divide by two on both sides.

### Example Question #1 : Solving Inequalities

Solve the inequality:

Explanation:

Distribute the negative through the terms of the binomial.

Subtract  on both sides.

Divide by 13 on both sides.

### Example Question #1 : Solving Inequalities

Solve

Explanation:

The first thing we can do is distribute the  and the  into their respective terms:

Now we can start to simplify by gathering like terms.  Remember, if we multiply or divide by a negative number, we change the direction of the inequality:

Solve  .