Solve the inequality. Graph and check your solutions.
4/3 <=n + 3
To solve the inequality, we can start by subtracting 3 from both sides:
4/3 - 3 <= n
To find a common denominator, we can multiply 3 by both sides:
(4/3 - 9/3) <= n
-5/3 <= n
Now we can graph the solution on the number line. To do this, we plot a closed circle at -5/3 and shade to the right:
<=======●---------------
Finally, we can check the solutions by plugging in a value within and outside the shaded region. Let's try n = -2:
4/3 <= -2 + 3
4/3 <= 1
Since 4/3 is less than or equal to 1, the inequality holds true for n = -2.
Now let's try n = 0:
4/3 <= 0 + 3
4/3 <= 3 1/3
Since 4/3 is less than or equal to 3 1/3, the inequality still holds true for n = 0 as well.
Thus, our graph and solutions are accurate.
To solve the inequality 4/3 <= n + 3, we need to isolate the variable n. Here are the steps:
Step 1: Subtract 3 from both sides of the inequality.
4/3 - 3 <= n + 3 - 3 simplifies to -2 2/3 <= n.
Step 2: Simplify the left side of the inequality.
-2 2/3 can be written as -8/3.
So, the simplified inequality is -8/3 <= n.
To graph this inequality, we draw a number line and plot a closed circle at -8/3 (since it includes equal to) and then shade the line to the right of -8/3 to represent all the solutions.
Finally, to check the solutions, we can pick arbitrary values for n, such as -3, -2, 0, 1, and 2, and substitute them into the inequality. If the inequality is true for these values, then they are indeed solutions.
For example:
-8/3 <= -3 + 3 simplifies to -8/3 <= 0, which is true.
-8/3 <= -2 + 3 simplifies to -8/3 <= 1/3, which is true.
-8/3 <= 0 + 3 simplifies to -8/3 <= 3, which is true.
-8/3 <= 1 + 3 simplifies to -8/3 <= 4, which is true.
-8/3 <= 2 + 3 simplifies to -8/3 <= 5, which is true.
Therefore, the solution to the inequality is n >= -8/3.
To solve the inequality 4/3 ≤ n + 3, we need to isolate the variable n on one side of the inequality sign.
1. Subtract 3 from both sides of the inequality:
4/3 - 3 ≤ n + 3 - 3
4/3 - 9/3 ≤ n
-5/3 ≤ n
So, the solution to the inequality is -5/3 ≤ n.
To graph the solution, we can represent it on a number line:
-∞ -5/3 ∞
+--+--+--+--+--[===========>]--+--+--+--+--+
-5/3
The open circle on -5/3 indicates that it is not inclusive, which means n can be equal to -5/3 but also any value greater.
To check the solution, substitute a value greater than or equal to -5/3 into the original inequality and verify if it holds true. For example, if we choose n = -5/3, we have:
4/3 ≤ -5/3 + 3
4/3 ≤ (-5 + 9)/3
4/3 ≤ 4/3
Since 4/3 is indeed less than or equal to 4/3, the solution is verified.