2/5^(5x-1) >orequal 25/4
Start with = then we will worry about the inequality.
2/5 raised to the 5x-1 = 25/4
take the log of both sides and then use the law of logs.
(5x-1)(log 2/5) = log (25/4)
(5x-1) = log(25/4) divided by (log 2/5)
simplify that then solve for x by adding 1 to both sides and then dividing by5.
Greater than will give your answer to the right on the number line
2/5^(5x-1) >= 25/4
8 >= 5^2*5^(5x-1)
8 >= 5^(5x+1)
5x+1 <= log8/log5
x <= (log8/log5 - 1)/5
To solve the inequality 2/5^(5x-1) ≥ 25/4, we need to isolate the variable (x) on one side of the inequality symbol. Here's how to do it step by step:
Step 1: Simplify both sides of the inequality if possible.
On the left side, we have 2/5^(5x-1). To simplify this, we can express 5^(5x-1) as (5^5)^x * 5^(-1) = 3125^x * 1/5^1 = 3125^x/5.
So the inequality now becomes 2/(3125^x/5) ≥ 25/4.
Step 2: Multiply both sides of the inequality by 4 to eliminate the fraction on the right side.
This gives us 8/(3125^x/5) ≥ 25.
Step 3: Invert the fraction on the left side of the inequality.
The inequality becomes (3125^x/5)/8 ≤ 1/25.
Step 4: Multiply both sides of the inequality by 8.
This results in (3125^x/5) ≤ 8/25.
Step 5: Multiply both sides of the inequality by 5.
The inequality becomes 3125^x ≤ 8/25 * 5.
Step 6: Simplify the right side of the inequality.
8/25 * 5 = 8/5 = 1.6.
Step 7: Take the logarithm of both sides of the inequality to eliminate the exponent.
Taking the logarithm (base 3125) on both sides we get x ≤ log3125(1.6).
Step 8: Use a calculator or software to evaluate the logarithm.
Log3125(1.6) ≈ -0.1739.
Therefore, the solution to the inequality 2/5^(5x-1) ≥ 25/4 is x ≤ -0.1739.