Answer:
(c) x < -5.170
Explanation:
The given function is an exponential function with a decay factor of 0.5 and a y-intercept of 0.5^2 = 0.25. It will be larger for more negative x-values.
Estimate
We know that 2^3 = 8, so (1/2)^-3 = 8. This means the exponent of 0.5 in the given inequality will be more negative than -3:
x +2 < -3
x < -5 . . . . . . . . subtract 2
The only answer choice in this range is ...
x < -5.170
Exact solution
Taking the logarithm of both sides of the inequality, we have ...
(x +2)log(0.5) > log(9)
x +2 < log(9)/log(0.5) . . . . . . log(0.5) < 0, so the inequality reverses
x < log(9)/log(0.5) -2 . . . . . . subtract 2
x < (0.95424251)/(-0.30103000) -2 = -3.1699250 -2
x < -5.1699250 ≈ -5.170