Refer to the example below.

a. Why is the solution set the empty set in Part C?

b. Why is the solution set $\{x \mid x>1\}$ in Part D?

Example:

Solve. Write the solution in set notation. Graph the solution.

C.

$$\text { C } \begin{array}{rlrl} 2 x-3 & >3 \text { AND } x+4 & \leq 1 \\ 2 x & >6 & x & \leq-3 \\ x & >3 & \end{array}$$

Solution set: $\{x \mid x>\quad\} \cap\{x \mid x \leq \quad\}$ or the ________ set or

D.

$$\begin{aligned} & \text {} 3 x-1>2 \text { OR } 2 x+2 \geq 8 \\ & 3 x>3 \quad 2 x \geq 6 \\ & x>1 \quad x \geq 3 \\ & \end{aligned}$$

Solution set: $\{x \mid x>\quad\} \cup\{x \mid x \geq \quad\}$ or $\{x \mid x>\quad\}$

Indicate whether the value of the variable, p = 40, is a solution of the inequality 4p < 64. Display your reasoning.

A carpenter cuts boards that are 2 meters long. It is acceptable for the length to vary from this value by at most 0.05 meters. Give and solve an absolute-value inequality to determine the range of acceptable lengths.

Which is the next and most efficient step in solving the inequality for x?

-4x > 12

A. $\frac{-4 x}{-4}<\frac{12}{-4}$

B. $\frac{-4 x}{4}>\frac{12}{4}$

C. $\frac{-4 x}{-4}>\frac{12}{-4}$

D. $\frac{-4 x}{4}<\frac{12}{4}$

Refer to the example below.

a. Can Kristin spend $40 on the meal before the tip? Why or why not?

b. What whole dollar amount is the most Kristin can spend before the tip? Justify.

c. The solution set of an equation or inequality consists of all values that make the statement true. Explain the whole dollar amounts that are in the solution set for this situation.

d. Assume Kristin also has to pay a 6% meal tax. Give an inequality to represent the new situation. Then identify two solutions.

Example

Kristin can afford to spend at most$50 for a birthday dinner at a restaurant, including a 15% tip. Describe some costs that are within her budget.

A Which inequality symbol can be used to represent "at most"?

B Complete the verbal model for the situation.

Cost before tip (dollars) 15% Cost before tip (dollars) Budget limit (dollars)

C Write and simplify an inequality for the model.

D Complete the table to find some costs that are within Kristin's budget.

$$\begin{array}{|c|c|c|c|} \hline \text { Cost } & \text { Substitute } & \text { Compare } & \text { Solution? } \\ \hline \$ 47 & 1.15(47) \leq 50 & 54.05 \leq 50 \times & \text { No } \\ \hline \$ 45 & 1.15(45) \leq 50 & & \\ \hline \$ 43 & & & \\ \hline \$ 41 & & & \\ \hline \end{array}$$

How can you tell the difference between the addition and subtraction properties of inequality and the addition and subtraction properties of equality?

Indicate whether the value, $m=8 ; 2 m \geq 6$, is a solution to the inequality.

The Mexican Tetra is a tropical fish that needs a water temperature between 68 and 77 degrees Fahrenheit, inclusive. An aquarium is heated 8 degrees so that a Tetra can live in it. What temperatures could the water have been before the heating? Give and solve an inequality for the situation.

Determine the solution set. Explain each step and graph the solution set.

$$12-3 x \leq 6$$

Solve 4x < 32. Explain your steps. Write your solution using set notation. Graph your solution.

Refer to the example below.

a. Explain why the Distributive Property applies first in the solution.

b. Could the properties have been applied in a different order than displayed below? If so, would this make finding the solution easier or more difficult? Discuss.

c. How would the solution change if the simplified coefficient of x were negative?

Example:

Find the solution set. Justify each step and graph your solution.

$$\begin{array}{rlrl} 3(2 x-3) & \leq x+1 & \\ 6 x-9 & \leq x+1 & & \\ 6 x \quad\quad -9 & \leq x \quad\quad& +1& \\ & \leq & & \\ 5 x-9+9 & \leq & & \text { Simplify. } \\ & \leq & & \\ & \leq & & \\ & \leq & & \text { Simplify. } \end{array}$$

Solution set: _________

Sumiko is allowed to watch no more than 10 hours of television every week. She has watched 4 hours of television already. Give and solve an inequality to display how many more hours of television Sumiko can watch.

Solve the inequality, -3a + 10 < -11, and graph the solution.

Solve and write the solution in set notation. Graph your solution.

$4 x+2>14$ AND $x+6 \leq 4$