Question

1. The solution of the inequality $\left|x\right|\mathbf{\le }\mathbf{3}$is the interval$\_\_\_\_\_\_\_\_\_$.
2. The solution of the inequality $\left|x\right|\mathbf{\ge }\mathbf{3}$is a union of two intervals $\_\_\_\cup \_\_\_$.

Short answer

1. The interval notation is $x\in \left[-3,3\right].$
   
2. The required answer is$x\in \left(-\infty ,\left.3\right]\right.\cup \left[3,\left.\infty \right)\right..$
   

Step-by-step solution

Part a. Step 1. Consider the given inequality

The given inequality is $\left|x\right|\le 3$.

Part a. Step 2. Interpret the given inequality.

With the absolute value as the distance from 0, we find that any number within units will satisfy the inequality.

Part a. Step 3. Satisfy the equation numerically.

Since,$-3\le x\le 3$ in the interval notation, we write this with square brackets. Since, the endpoints$-3$ and$3$ are included in the interval.

Thus, the interval is given by$x\in \left[-3,3\right].$

Part b. Step 1. Consider the given inequality.

The given inequality $\left|x\right|\ge 3$is absolute value as the distance from 0, thus any number that is more than 3 units from 0 will satisfy the inequality. In the positive direction, this is given by the following inequality, $x\ge 3$.

In the negative direction, this is given by any satisfying the following inequality,$x\le -3.$

Part b. Step 2. Write the inequality interval notation.

We use square brackets since the inequalities are strict. Thus, the interval notation for $x\ge 3$is given $x\in \left[3,\infty \right]$.

The interval notation for the equation$x\le -3$ is given$x\in \left(-\infty ,\left.-3\right].\right.$

Part b. Step 3. Combined the obtained interval.

Writing these two intervals as a union, we get,$x\in \left(-\infty ,\left.3\right]\right.U\left[3,\left.\infty \right)\right..$