Question

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

1. $b+\frac{7}{8}\ge \frac{1}{6}$
   
2. $6y<48$
3. $40<\frac{5}{8}k$
   

Short answer

(a) The solution interval for inequality (a) is $[-\frac{17}{24},\infty )$and representation on number line is :-

(b)The solution interval for inequality (b) is $\left(-\infty ,8\right)$and representation on number line is :-

(c) The solution interval for inequality (c) is $\left(64,\infty \right)$and representation on number line is :-

Step-by-step solution

Step 1. Part (a) Given and Solution

We have given the following inequality :-

$b+\frac{7}{8}\ge \frac{1}{6}$

To solve the inequality subtract $\frac{7}{8}$from both sides, then we have :-

$$\begin{gathered} b+\frac{7}{8}-\frac{7}{8}\ge \frac{1}{6}-\frac{7}{8} \\ \Rightarrow b\ge -\frac{17}{24} \end{gathered}$$

So the solution interval is $[-\frac{17}{24},\infty )$.

Representation of solution on number line is as following :-

Step 2. Part (b) Given and solution

Here we have the following inequality :-

$6y<48$.

To solve this inequality divide both sides by 6, then we have :-

$$\begin{gathered} \frac{6y}{6}<\frac{48}{6} \\ \Rightarrow y<8 \end{gathered}$$

So the solution interval for this inequality is :-

$\left(-\infty ,8\right)$.

Representation of solution on number line is as following :-

Step 3. Part (c) Given and Solution

Now we have following inequality :-

$40<\frac{5}{8}k$

To solve the inequality multiply both sides by $\frac{8}{5}$, then we have :-

$$\begin{gathered} 40*\frac{8}{5}<\frac{5}{8}k*\frac{8}{5} \\ \Rightarrow 64<k \\ ork>64 \end{gathered}$$

So the solution interval is $\left(64,\infty \right)$.

Representation of solution on number line is as following :-