Chapter 2.6, Problem 38E

To determine

To write:an absolute value inequality for the given graphs

Answer to Problem 38E

Inequality for first graph is $|x-2|\ge 3$

Inequality for second graph is $|x-2|<3$

Inequality for third graph is $|x-2|\le 3$

Inequality for fourth graph is $|x-2|>3$

Explanation of Solution

Given:

  

Calculation:

Write an absolute value inequality for the first graph:

First chose the" $>\text{"}$ and $\ge \text{ "}$ signs because thesolution set is outside the interval where the meannumber (2) lies. From the two left signs, chose the $\text{"}\ge \text{ "}$ sign because the boundary points (-1,5) are included in the solution set.

  $|x-2|\ge 3$

Write an absolute value inequality for the second graph:

First chose the" $<\text{"}$ and $\le \text{ "}$ signs because the solution set is inside the interval where the meannumber (2) lies. From the two left signs, chose the " $<\text{"}$ sign because the boundary points (-1,5) are not included in the solution set.

  $|x-2|<3$

Write an absolute value inequality for the third graph:

First chose the" $<\text{"}$ and $\le \text{ "}$ signs because the solution set is inside the interval where the meannumber (2) lies. From the two left signs, chose the $\text{"}\le \text{ "}$ sign because the boundary points (-1,5) are included in the solution set.

  $|x-2|\le 3$

Write an absolute value inequality for the fourth graph:

First chose the" $>\text{"}$ and $\ge \text{ "}$ signs because thesolution set is outside the interval where the mean number (2) lies. From the two left signs, chose the $\text{"}>\text{"}$ sign because the boundary points (-1,5) are not included in the solution set.

  $|x-2|>3$

Conclusion:

Therefore,

Inequality for first graph is $|x-2|\ge 3$

Inequality for second graph is $|x-2|<3$

Inequality for third graph is $|x-2|\le 3$

Inequality for fourth graph is $|x-2|>3$