Solve the inequality. Graph and check your solutions.
-1/2<r-1/2
To solve the inequality, we will first isolate the variable r on one side of the inequality.
-1/2 < r - 1/2
Adding 1/2 to both sides, we get:
-1/2 + 1/2 < r - 1/2 + 1/2
0 < r
So the solution to the inequality is r > 0.
Now let's graph this inequality on the number line:
0
We shade the region to the right of 0 to represent r > 0.
Finally, let's check a value in that region to see if it satisfies the inequality. Let's choose r = 1.
-1/2 < 1 - 1/2
-1/2 < 1/2 (which is true)
Since the inequality holds for r = 1, we can conclude that the solution to the inequality is r > 0.
To solve the inequality -1/2 < r - 1/2, we will go through the following steps:
Step 1: Add 1/2 to both sides of the inequality to isolate the variable r:
-1/2 + 1/2 < r - 1/2 + 1/2
0 < r
Step 2: Simplify the inequality:
r > 0
Now we will graph the solution on a number line. Since the inequality is r > 0, the solution includes all real numbers greater than the number 0. We represent this by shading all the points to the right of 0 on the number line.
Number Line:
-3 -2 -1 0 1 2 3
-------------------------------------------
+ x
In this graph, the point 0 is represented by a filled-in circle, indicating that it is not part of the solution. The arrow to the right illustrates that the solution includes all the points greater than 0.
To check the solution, you can substitute any value greater than 0 into the original inequality and see if it holds true.
For example, let's check r = 1:
-1/2 < 1 - 1/2
-1/2 < 1/2
This inequality is true, so the solution is valid.
Similarly, if we check a value smaller than 0, let's say, r = -1:
-1/2 < -1 - 1/2
-1/2 < -3/2
This inequality is not true, so the value -1 is not a part of the solution.
Therefore, the solution to the inequality -1/2 < r - 1/2 is r > 0.
To solve the inequality -1/2 < r - 1/2, we can follow these steps:
Step 1: Add 1/2 to both sides of the inequality:
-1/2 + 1/2 < r - 1/2 + 1/2
0 < r
Step 2: The solution is r > 0.
We can graph this inequality on a number line as follows:
------------------------o------------>
-3 -2 -1 0 1 2 3
Since the inequality is r > 0, we graph an open circle at 0 and an arrow pointing to the right.
To check the solution, we can select any value greater than 0 (let's say 1) and substitute it back into the original inequality.
-1/2 < 1 - 1/2
-1/2 < 1/2
The inequality holds true, so the solution r > 0 is correct.
Note: If the inequality had been less than or equal to (-1/2 ≤ r - 1/2), we would have graphed a closed circle at 0 instead of an open circle since it includes the endpoint.