Chapter 3.6, Problem 23PPE

To determine

To find: The interval as an inequality and graph the solution.

Answer to Problem 23PPE

To graph the solution, draw a number line and indicate the solution on it.

Closed circle denoted that $2$ is included. The shaded part denoted the solution.

Explanation of Solution

Given:

  $\left(-\infty ,2\right]$ .

Concept used:

The difference is that a solution to the inequality is not the drawn line but area of the coordinate plane that satisfy the inequality.

The common area is the solution of inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

A compound inequality is a combination of two or more inequalities joined by either an “and” or an “or”. In mathematics, most often it deals with equalities two statements that must, at all times, remain exactly equal to each other, however, sometimes interested in more than or less than relationship.

There is a statement and one must always be either bigger than or at least as big as the other.

Calculation:

The given interval $\left(-\infty ,2\right]$ denotes all real numbers less than or equal to $2$ ,since the bracket indicated that $2$ is included.

Let $x$ denote the solution for the given interval.

Then the given interval notation can be written in inequality as:

  $x\le 2$ .

Hence, to graph the solution, draw a number line and indicate the solution on it.

Closed circle denoted that $2$ is included. The shaded part denoted the solution.