Solve the following inequality and write your answer using interval notation.
Please show all of your work.
(1 / (x+8) ) ≤ ((8-5x)/43)
To solve the inequality (1 / (x+8) ) ≤ ((8-5x)/43), we will follow these steps:
Step 1: Eliminate the fractions by multiplying both sides of the inequality by the LCM (Least Common Multiple) of the denominators.
The LCM of (x + 8) and 43 is 43(x + 8). So, we will multiply both sides by 43(x + 8):
43(x + 8) * (1 / (x + 8)) ≤ 43(x + 8) * ((8 - 5x) / 43)
Simplifying the expression:
43 * (x + 8) * (1 / (x + 8)) ≤ (8 - 5x)
43 cancels out with 43, and (x + 8) cancels out with (x + 8):
1 ≤ (8 - 5x)
Now we have: 1 ≤ 8 - 5x.
Step 2: Simplify the inequality by combining like terms:
1 ≤ 8 - 5x
Subtract 8 from both sides:
1 - 8 ≤ 8 - 8 - 5x
-7 ≤ -5x
Step 3: Divide both sides of the inequality by -5. When dividing by a negative number, we need to change the direction of the inequality:
(-7) / (-5) ≥ (-5x) / (-5)
7/5 ≥ x
Step 4: Write the solution in interval notation:
The solution to the inequality is x ≥ 7/5, which can be written in interval notation as [7/5, ∞).
Note: In interval notation, square brackets [ ] indicate "including," and the symbol ∞ represents infinity, indicating that the solution includes all values greater than or equal to 7/5.