Solving and Graphing One Variable Absolute Value Inequalities

We are asked to graph the solution set for all possible values of x in the given inequality. |x|<3 To do this, we will create a compound inequality by removing the absolute value. In this case, the inequality represents the set of all numbers that are less than 3 and greater than - 3.

The first inequality tells us that all values greater than -3 will satisfy the inequality. The second inequality tells us that all values less than 3 will satisfy the inequality. The intersection of these two solution sets is the solution set of the compound inequality. First Solution Set:& -3 < x Second Solution Set:& x < 3 Intersecting Solution Set:& -3 < x < 3 The graph of this inequality includes all values from -3 to 3, not inclusive. We can show this by using open circles on the endpoints.

This corresponds to option C.

We are asked to graph the solution set for all possible values of y in the given inequality. |y|≥ 4.5 We can rewrite this inequality as a compound inequality.

The first inequality tells us that all values less than or equal to - 4.5 will satisfy the inequality. The second inequality tells us that all values greater than or equal to 4.5 will satisfy the inequality. The union of these two solution sets is the solution set of the compound inequality. First Solution Set: & y≥ 4.5 Second Solution Set: & y≤ - 4.5 Combined Solution Set: & y≤ - 4.5 or y≥ 4.5 The graph of this inequality includes all values less than or equal to - 4.5 or greater than or equal to 4.5. We can show this by keeping the endpoints closed.

This corresponds to option D.

We are asked to find the solution set of the given absolute value inequality. 3|14-m|>18 Let's start by isolating the absolute value expression.

This inequality means that the distance between m and 14 is greater than 6. We can write it as a compound inequality. 14-m < - 6 or 14-m > 6 We can solve the individual inequalities by performing inverse operations on both sides of the inequality. Let's first solve 14-m < - 6.

This inequality tells us that all values greater than 20 will satisfy the inequality. Now, we will solve the other inequality.

This inequality tells us that all values less than 8 will satisfy the inequality. The combination of the solution sets for the individual inequalities is the solution set of the given absolute value inequality. First Solution Set:& m< 8 Second Solution Set:& m>20 Combined Solution Set:& m< 8 or m >20

We will now graph the absolute value inequality. The graph of the given inequality includes all values less than 8 or greater than 20. Since the inequality sign is strict, the endpoints are not included. Therefore, the endpoints should be open circles.

This corresponds to C.

We have been told that essay contest entries can have 500 words with an absolute deviation of at most 40 words. Let w be the number of words written. The absolute deviation is the difference between w and 500. We can state this as an absolute value inequality as follows. |w- 500|≤ 40 Another way of looking at this is to say an essay that falls under 40 words below 500 will not be accepted, and an essay with 40 words over 500 also will not be accepted.

We need to solve the absolute value inequality written in Part A to find the minimum and maximum acceptable numbers of words. To do this, we will rewrite the inequality as a compound inequality. Absolute Value Inequality |w-500|≤ 40 ⇓ Compound Inequality - 40 ≤ w-500 ≤ 40 This compound inequality means that the distance from w-500 is greater than or equal to - 40 and less than or equal to 40. w-500≥- 40 and w-500≤ 40 We can solve the individual inequalities by performing inverse operations. Let's first solve w-500 ≥ - 40.

This inequality tells us that all values greater than or equal to 460 will satisfy the inequality. Now, let's solve the other inequality.

This inequality tells us that all values less than or equal to 540 will satisfy the inequality. The solution to this type of compound inequality is the union of the solution sets. First Solution Set:& 460≤ w Second Solution Set:& w≤ 540 Intersecting Solution Set:& 460≤ w≤ 540 Therefore, the minimum acceptable number of words is 460 and the maximum acceptable number of words is 540.

We know that most pet birds’ comfort range is between 65 and 80 degrees Fahrenheit. Let's show this range on a number line.

To write an absolute value inequality representing this range, we need to find the midpoint between 65 and 80. Let d be the distance to the midpoint. To find d, we will find the half of the difference between the endpoints.

The distance d to the midpoint from the endpoints is 7.5. Therefore, the midpoint is equal to 65+7.5 = 72.5.

The points that are within 7.5 units of the midpoint represent the range and they can be written as the following absolute value inequality. Since the endpoints are included, the inequality should be non-strict. |x- 72.5| ≤ 7.5

Let's examine the steps.

To solve absolute value inequalities we must write the given inequality as two inequalities. If we have an inequality of the form |x+a|non-negative number, we can write it as a compound inequality. Think of this as a compound sentence, in this case, that uses and. x+a < b and x+a>- b Notice that in the exercise of the student's work provided the inequality of the form x+a > - b is not considered. Given what was just explained about compund inequalities, the student should have also solved x-5>- 20. Therefore, we can conclude that the inequality was not written correctly as a compound inequality.

Let's solve the inequality by correcting this mistake!

Here, we only applied Addition Property of Inequality to solve the individual inequalities. The solution set of the absolute value inequality is - 15 < x < 25.

Before we can write the inequality for the given graph, we should remember two points.

1. An absolute value equation is written as |x-a|=b, where a is the midpoint on the number line and b is distance from the midpoint to either endpoint.
2. The midpoint can be found as the mean of the two endpoints P_1 and P_2.

Midpoint=P_1+P_2/2 Consider the given graph.

We can see that the endpoints are -5 and 1. Knowing this we can find the midpoint.

The midpoint at - 2 is at a distance of 3 from -5 and 1. Therefore, we can write the absolute value equation. |x-( - 2)|= 3 ⇔ |x+2| = 3 In order to write it as an absolute value inequality, we first notice in the graph that the endpoints are closed. This means that the inequality is non-strict. Since the solution set includes points that are farther away from the endpoints, the distance is greater than or equal to 3. |x+2|≥ 3

Consider the given graph.

Since this graph has the same endpoints, the midpoint is also - 2. We can write the same absolute value equation. |x+2|=3 In order to write it as an absolute value inequality, we first notice in the graph that the endpoints are open. This means that the inequality is strict. Since the solution set includes points that are between the endpoints, the distance is less than 3. |x+2|<3

Consider the given graph.

Since this graph has the same endpoints, the midpoint is also - 2. We can write the same absolute value equation. |x+2|=3 In order to write it as an absolute value inequality, we first notice in the graph that the endpoints are closed. This means that the inequality is non-strict. Since the solution set includes points that are between the endpoints, the distance is less than or equal to 3. |x+2|≤ 3

Consider the given graph.

Since this graph has the same endpoints, the midpoint is also - 2. We can write the same absolute value equation. |x+2|=3 In order to write it as an absolute value inequality, we first notice in the graph that the endpoints are open. This means that the inequality is strict. Since the solution set includes points that are farther away from the endpoints, the distance is greater than 3. |x+2|>3