Question

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

1. $a+\frac{3}{4}\ge \frac{7}{10}$
   
2. $8x>72$
   
3. $20>\frac{2}{5}h$
   

Short answer

(a) The solution interval for inequality (a) is $[-\frac{1}{20},\infty )$and representation on number line is :-

(b)The solution interval for inequality (b) is $\left(9,\infty \right)$and representation on number line is :-

(c) The solution interval for inequality (c) is $\left(-\infty ,50\right)$and representation on number line is :-

Step-by-step solution

Step 1. Part (a) Given and Solution

We have given the following inequality :-

$a+\frac{3}{4}\ge \frac{7}{10}$

Now to solve this inequality, subtract $\frac{3}{4}$from both sides, then we have :-

$$\begin{gathered} a+\frac{3}{4}-\frac{3}{4}\ge \frac{7}{10}-\frac{3}{4} \\ \Rightarrow a\ge -\frac{1}{20} \end{gathered}$$

So the solution interval for this inequality is :-

$[\frac{-1}{20},\infty )$.

Representation of solution on number line is as following :-

Step 2. Part (b) Given and solution

Here we have given the following inequality :-

$8x>72$.

To solve this inequality divide both sides by 8, then we have :-

$$\begin{gathered} \frac{8x}{8}>\frac{72}{8} \\ \Rightarrow x>9 \end{gathered}$$

So the solution interval is $\left(9,\infty \right)$.

Representation of solution on number line is as following :-

Step 3. Part (c) Given and solution

Now we have following inequality :-

$20>\frac{2}{5}h$.

To solve this inequality multiply both sides by $\frac{5}{2}$, then we have :-

$$\begin{gathered} 20*\frac{5}{2}>\frac{2}{5}h*\frac{5}{2} \\ \Rightarrow 50>h \\ orh<50 \end{gathered}$$

Then the solution interval is :-

$\left(-\infty ,50\right)$.

Representation of solution on number line is as following :-