How to solve inequality with variable on both sides.
With n+7≤3n−1 as an example (the question I am trying to solve) I ask that someone provide a step by step example of how to solve this, I've made multiple attempts and It always ends up botched. I think there is a step I don't know to use when there is a variable on both sides, so that would be very appreciated.
To solve an inequality with a variable on both sides, follow these step-by-step instructions:
Step 1: Simplify the inequality if necessary. Combine like terms on both sides of the inequality.
Example: n + 7 ≤ 3n - 1
Step 2: Get all the variable terms on one side by subtracting or adding them. To do this, you'll need to add or subtract the variable term that is on the opposite side of the inequality symbol.
Example: Subtract n from both sides:
n - n + 7 ≤ 3n - n - 1
7 ≤ 2n - 1
Step 3: Simplify the inequality again if necessary. Combine like terms on the side where the variable terms are now located.
Example: 7 ≤ 2n - 1
Step 4: Get the constant term on the opposite side by subtracting or adding it.
Example: Add 1 to both sides:
7 + 1 ≤ 2n - 1 + 1
8 ≤ 2n
Step 5: Divide both sides of the inequality by the coefficient of n to solve for n. Remember to flip the inequality symbol if you divide by a negative number.
Example: Divide both sides by 2 (coefficient of n):
8/2 ≤ (2n)/2
4 ≤ n
Step 6: Rearrange the inequality so that n is on the left side.
Example: n ≥ 4
So, the solution to the inequality n + 7 ≤ 3n - 1 is n ≥ 4.
To solve an inequality with a variable on both sides, such as "n+7≤3n−1," follow these steps:
1. Start by simplifying both sides of the inequality. In this example, you can simplify by combining like terms.
n + 7 ≤ 3n - 1
2. Next, isolate the variable on one side of the inequality. You can do this by moving all the terms involving the variable to one side and the constants to the other side.
Subtract "n" from both sides to move all the terms involving "n" to the left side:
n - n + 7 ≤ 3n - n - 1
Simplifying further:
7 ≤ 2n - 1
3. Rearrange the equation, if necessary, so that the variable term is on the left side and the constant term is on the right side:
2n - 1 ≥ 7
4. Solve for the variable by isolating it. In this case, you want to isolate "n" by moving the constant term to the opposite side of the inequality. Add "1" to both sides:
2n - 1 + 1 ≥ 7 + 1
Simplifying further:
2n ≥ 8
5. Finally, solve for the variable by dividing both sides of the inequality by the coefficient of "n" (in this case, 2):
2n/2 ≥ 8/2
Simplifying further:
n ≥ 4
The solution to the inequality n + 7 ≤ 3n - 1 is n ≥ 4. This means that any value of "n" greater than or equal to 4 will satisfy the inequality.