Question

Solve the inequality $9h-7\left(2-h\right)<8\left(h+11\right)+8h$, graph the solution on the number line, and write the solution in interval notation.

Short answer

The solution in interval notation is $\left(-\infty ,\infty \right)$

The solution on number line is:

Step-by-step solution

Step 1. Given Information

Given inequality is$9h-7\left(2-h\right)<8\left(h+11\right)+8h$

The inequality is to be solved, the solution is to be graphed on the number line, and the solution is to be written in interval notation.

Step 2. Solving the inequality

Isolating the variable in the inequality:

$$\begin{gathered} 9h-7\left(2-h\right)<8\left(h+11\right)+8h \\ 9h-14+7h<8h+88+8h \\ 16h-14<16h+88 \\ 16h-16h-14<16h-16h+88 \\ -14<88 \end{gathered}$$

Here the variable is gone and the statement is always true i.e. the solution is all real numbers.

Hence the solution in the interval notation is$\left(-\infty ,\infty \right)$

Step 3. Graphing the inequality

On a number line, if the inequality is less than or greater than then a hollow dot is used to indicate that the number is not included. If the inequality is less than equals or greater than equals then a solid dot is used to indicate that the number is included.

Graphing the interval on a number line: