Chapter 5.1, Problem 52HP

To determine

Compare and contrast the graph of $a<4\text{and}a\le 4$

Explanation of Solution

Given:

The inequality: $a<4\text{and}a\le 4$

Concept Used:

There are four different types of inequalities:

Greater than − $(>)$ ; Less than − $(<)$ ; Greater than or equal to − $(\ge )$ ; Less than or equal to − $(\le )$

• For = and <, the arrow points down to the left.

  For ≤ and >, the arrow points up to the right.

Representation of $a<4$ on the number line

An open circle is used for greater than (>) or less than (<). The point is not part of the solution. The graph then extends endlessly in one direction.

For < and >, use an open circle to indicate the number itself is not part of the solution.

This means all number less than 4, but 4 is not including for $a<4$

  

Representation of $a\le 4$ on the number line

A closed, or shaded, circle is used to represent the inequalities greater than or equal to ( $\ge$ ) or less than or equal to ( $\le$ ). The point is part of the solution.

For ≤ and ≥ , use a closed dot to indicate the number itself is part of the solution.

This means all numbers less than or equal to 4 is the solution for $a\le 4$

  

Thus, the graph shows for $a<4$ starts with an open circle at 4 and moving endlessly to the left of 4. This means all number less than 4, but 4 is not including for $a<4$ and the graph shows for $a\le 4$ starts with an closed circle at 4 and moving endlessly to the left of 4. This means all numbers less than or equal to 4 is the solution for $a\le 4$