Answer:
To solve the given inequality, let's break it down step by step:
a - 2/3 ≤ 2a ± 3/5 ± 5/8
First, simplify the expressions on both sides of the inequality:
a - 2/3 ≤ 2a ± 3/5 ± 5/8
Let's start with the right side:
2a ± 3/5 ± 5/8
To simplify this, we need a common denominator. The least common multiple of 5 and 8 is 40, so:
2a ± (3/5) ± (5/8) = (16a ± 24 ± 25)/40
Now we can rewrite the inequality:
a - 2/3 ≤ (16a ± 24 ± 25)/40
Next, let's get rid of the fractions by multiplying both sides of the inequality by 40:
40(a - 2/3) ≤ 16a ± 24 ± 25
Expanding:
40a - 80/3 ≤ 16a ± 24 ± 25
Now, let's simplify the expression on the left side:
40a - 80/3 = (120a - 80)/3
The inequality becomes:
(120a - 80)/3 ≤ 16a ± 24 ± 25
To eliminate the fraction, we can multiply both sides of the inequality by 3:
3(120a - 80)/3 ≤ 3(16a ± 24 ± 25)
Simplifying:
120a - 80 ≤ 48a ± 72 ± 75
Now, let's combine like terms:
120a - 80 ≤ 48a ± 147
Next, let's isolate the variable by moving the terms containing 'a' to one side:
120a - 48a ≤ 147 + 80
72a ≤ 227
Finally, divide both sides by 72 to solve for 'a':
a ≤ 227/72
The solution to the inequality is a ≤ 227/72.