Recall that with absolute values and "less than" inequalities, we have to hold the following:

12x + 3y < 15

AND

12x + 3y > –15

Otherwise written, this is:

–15 < 12x + 3y < 15

In this form, we can solve for y. First, we have to subtract x from all 3 parts of the inequality:

–15 – 12x < 3y < 15 – 12x

Now, we have to divide each element by 3:

(–15 – 12x)/3 < y < (15 – 12x)/3

This simplifies to:

–5 – 4x < y < 5 – 4x

This inequality could be rewritten as:

4x + 14 > 30 OR 4x + 14 < –30

Solve each for x:

4x + 14 > 30; 4x > 16; x > 4

4x + 14 < –30; 4x < –44; x < –11

Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.

For this problem, we must take into account the absolute value.

First, we solve for 2x– 2 > 20. But we must also solve for 2x– 2 < –20 (please notice that we negate 20 and we also flip the inequality sign).

First step:

2x– 2 > 20

2x> 22

x> 11

Second step:

2x– 2 < –20

2x< –18

x< –9

Therefore,x> 11 andx< –9.

A possible value forxwould be –10 since that is less than –9.

Note: the value 11 would not be a possible value forxbecause the inequality sign given does not include an equal sign.

Move +5 using subtraction rule which will give you.

Divide both sides by 2 (using division rule) and you will getwhich is the same as

Subtract 5 from both sides of the inequality:

Multiply both sides by 5:

Therefore only I must be true.

Solve for bothx– 3 < 2 and –(x– 3) < 2.

x– 3 < 2 and –x+ 3 < 2

x< 2 + 3 and –x< 2 – 3

x< 5 and –x< –1

x< 5 andx> 1

The results arex< 5 andx> 1.

Combine the two inequalities to get 1 <x< 5

Manipulate the inequality untilis on a side by itself:

For this equation,must be less than 6. Find the answer choice with values all less than 6. In this case, it will be -1, 4, and 5.

Subtractfrom both sides:

Subtract 7 from both sides:

Divide both sides by:

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

Sinceis not an answer, we must find an answer that, at the very least, does not contradict the fact thatis 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 thatis less than 5.

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

and:

Notice that any positive value forwsatisfies the second inequality above. Sincewmust be between 356 and 364, the first inequality above is the only reasonable choice.