Question

1. multi-step use the following inequality to answer the following questions. -5(x - 2) < 2x + 8 + 3x mp 7

a. which statements are true?
a the solution set is the empty set.
b the solution set is all real numbers.
c -5x + 10 < 5x + 8 is an equivalent inequality.
d 5x - 10 < 5x + 8 is an equivalent inequality.
e the number line is partially shaded.
b. suppose the solution is restricted to negative numbers. what is the solution?
c. suppose the solution is restricted to integers only. graph the solution on a number line.

1. it costs $30 to rent a moving van plus $0.75 per mile. what is the maximum distance that can be driven to keep the total cost below $50? mp 4,7

a 20
b 24
c 26
d 27

Explanation:

Step1: Simplify the inequality

$$\begin{align*} -5(x - 2)&<2x + 8+3x\\ -5x+10&<5x + 8 \end{align*}$$

So option C is correct.

Step2: Solve the inequality

$$\begin{align*} -5x+10&<5x + 8\\ -5x-5x&<8 - 10\\ -10x&<-2\\ x&>\frac{-2}{-10}\\ x&>\frac{1}{5} \end{align*}$$

Since \(x>\frac{1}{5}\), the solution set is not the empty set (A is wrong), not all real numbers (B is wrong), and the number line is not partially shaded (E is wrong).

Step3: For part b

If we restrict to negative numbers (\(x<0\)), and we know \(x>\frac{1}{5}\), the intersection of \(x < 0\) and \(x>\frac{1}{5}\) is the empty set.

Step4: For part c

Since \(x>\frac{1}{5}\), and restricting to integers, we start from \(x = 1\). On the number - line, we draw an open circle at \(x=\frac{1}{5}\) (but since we are restricting to integers, we can also think of it as starting from the first integer greater than \(\frac{1}{5}\) which is \(x = 1\)) and shade to the right.

Answer:

a. C
b. The empty set
c. On the number - line, draw an open circle at \(x=\frac{1}{5}\) (or start shading from \(x = 1\) when considering integers) and shade to the right.