Break up the statement into parts.

Two-fifths of a number:  $\displaystyle \frac{2}{5}x$

Eight less than two-fifths of a number:  $\displaystyle \frac{2}{5}x-8$

Must exceed seven:  $\displaystyle >7$

The answer is:  $\displaystyle \frac{2}{5}x-8>7$

Evaluate the inner quantity first.

Three less a number:  $\displaystyle x-3$

The squared quantity of three less a number:  $\displaystyle (x-3)^2$

Five less than the squared quantity of three less a number:  $\displaystyle (x-3)^2-5$

Must be more than eight:  $\displaystyle >8$

Combine the parts to set up the inequality.

The answer is:  $\displaystyle (x-3)^2-5>8$

Break up the sentence into parts.

A number squared:  $\displaystyle x^2$

Seven times a number squared:  $\displaystyle 7x^2$

Eight less than seven times a number squared:  $\displaystyle 7x^2-8$

Is at most four:  $\displaystyle \leq4$

Combine the parts to form the inequality.

The answer is:  $\displaystyle 7x^2-8\leq4$

Break up the inequality into parts.

Twice a number squared:  $\displaystyle 2x^2$

Ten less than twice a number squared:  $\displaystyle 2x^2-10$

Exceeds five:  $\displaystyle >5$

The answer is:  $\displaystyle 2x^2-10>5$

Break up the sentence into parts.

Three times a number:  $\displaystyle 3x$

Seven less than three times a number:  $\displaystyle 3x-7$

Four times the quantity of seven less than three times a number:  $\displaystyle 4(3x-7)$

Cannot exceed eight:  $\displaystyle \leq8$

Combine the parts to form the inequality.

The answer is:  $\displaystyle 4(3x-7)\leq8$

In order to set up the equation for this inequality, break down the sentence into parts.  

Twice a number $\displaystyle x$: $\displaystyle 2x$

Is less than:  $\displaystyle <$

The difference of four and the number x :  $\displaystyle 4-x$

Twice the difference of four and the number:  $\displaystyle 2(4-x)$

Combine all the terms and signs.

The answer is:  $\displaystyle 2x< 2(4-x)$.

$\displaystyle \left | y+9 \right | > 5$

can be written as the compound inequality statement

$\displaystyle y+9 < -5 \textup{ or } y + 9 > 5$

In each expression, 9 can be subtracted to yield the inequality statement

$\displaystyle y < -14 \textup{ or } y > -4$

This is the correct response.

$\displaystyle \left | x - 7 \right | < 16$

is equivalent to the three-way inequality

 $\displaystyle -16 < x - 7 < 16$

Add 7 to all three expressions to yield the statement:

$\displaystyle -9 < x < 23$

This is the correct response.

Subtract 1 on both sides of the inequality to get

$\displaystyle y \leq 3x+4$

From here we plug in each set of coordinates to see which could satisfy the statement.

$\displaystyle \text{For } (0,2):$

$\displaystyle 2\le3(0)+4$

$\displaystyle 2\le4$

This is a true statement therefore we say that (0,2) satisfies the inequality.

To simplify an inequality we want to isolate the variable on one side of the inequality sign. In order to accomplish this remember to do the reverse opperation to move numbers from one side to the other.

$\displaystyle 3x-3\geq 6$

The reverse operation of subtraction is addition. Therefore to move the three from the left hand side to the right hand side we will need to add 3 on each side of the inequality to get

$\displaystyle 3x \geq 9$.

From here divide each side of the inequality by 3 to isolate the variable, and since 3>0 we get

$\displaystyle x\geq3$.