Question

Solve each inequality. Then graph it on a number line.

$12\le t-9$

Short answer

The required value of x is $\left[21,\infty \right)$. The graph on the number line is,

Step-by-step solution

Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < or >, use the open parenthesis or a hollow circle at that endpoint.

To graph the endpoint with inequality symbol $\le \text{or}\ge$, use the bracket or a solid circle at that endpoint.

Properties of inequality:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ 2.b\le c\Rightarrow ab\le ac,\text{if}a>0 \\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0 \end{gathered}$$

Step 2. Solve the inequality for x.

In order to solve the inequality:

$12\le t-9$

Add 9 both sides of the inequality, as

$$\begin{gathered} 12\le t-9 \\ 12+9\le t-9+9 \\ 21\le t \end{gathered}$$

Thus, the solution of the given inequality is every number greater than or equals to 21, this can be represented in the interval form, as $\left[21,\infty \right)$.

Step 3. Plot the graph on the number line.

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point 21 is not strict, so use a solid circle at point 21. And the solution interval at right endpoint $\infty$, thus every point greater than 21 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: