# SAT Math : Inequalities

## Example Questions

### Example Question #2 : Inequalities

Given the inequality above, which of the following MUST be true?

Explanation:

Subtract  from both sides:

Subtract 7 from both sides:

Divide both sides by :

Remember to switch the inequality when dividing by a negative number:

Since is not an answer, we must find an answer that, at the very least, does not contradict the fact that  is less than (approximately) 4.67.  Since any number that is less than 4.67 is also less than any number that is bigger than 4.67, we can be sure that  is less than 5.

### Example Question #3 : Inequalities

A factory packs cereal boxes. Before sealing each box, a machine weighs it to ensure that it is no lighter than 356 grams and no heavier than 364 grams. If the box holds  grams of cereal, which inequality represents all allowable values of ?

Explanation:

The median weight of a box of cereal is 360 grams. This should be an allowable value of w. Substituting 360 for w into each answer choice, the only true results are:

and:

Notice that any positive value for w satisfies the second inequality above. Since w must be between 356 and 364, the first inequality above is the only reasonable choice.

### Example Question #11 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Divide  on both sides.

### Example Question #12 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Divide  on both sides. Remember to flip the sign.

### Example Question #13 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

Add  and subtract  on both sides.

Divide  on both sides.

### Example Question #14 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since its absolute value.

Divide  on both sides.

Divide  on both sides which flips the sign.

Divide  on both sides.

Since we have the 's being either greater than or less than the values, we can combine them to get .

### Example Question #15 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since it's absolute value.

Divide  on both sides.

Distribute the negative sign to each term in the parenthesis.

Add  and subtract  on both sides.

Divide  on both sides.

We must check each answer. Let's try .

This is not true therefore  is not correct. Let's try .

This true so therefore  is correct.

### Example Question #16 : Inequalities

Solve for .

Explanation:

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation.

We need to set-up two equations since it's absolute value.

Subtract  and add  on both sides.

Distribute the negative sign to each term in the parenthesis.

Divide  on both sides.

We must check each answer. Let's try .

This is not true therefore  is not correct. Let's try .

This true so therefore  is correct.

### Example Question #10 : Inequalities

What values of x make the following statement true?

|x – 3| < 9

x < 12

–6 < x < 12

6 < x < 12

–12 < x < 6

–3 < x < 9

–6 < x < 12

Explanation:

Solve the inequality by adding 3 to both sides to get x < 12.  Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.

### Example Question #1 : How To Find The Solution To An Inequality With Addition

If –1 < w < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?

|w|0.5

w/2

3w/2

|w|

w2