Use an over/under table to estimate the solution to the exponential equation 3+2^3x=9 to two decimal places.(1 point)
To estimate the solution to the equation 3 + 2^(3x) = 9, we can create an over/under table using different values of x.
Let's start by trying some values of x and see if the left side of the equation gets closer to, equal to, or farther away from the right side:
For x = -1:
3 + 2^(3(-1)) = 3 + 2^(-3) = 3 + 1/8 = 3.125 (under)
For x = 0:
3 + 2^(3(0)) = 3 + 2^0 = 3 + 1 = 4 (under)
For x = 1:
3 + 2^(3(1)) = 3 + 2^3 = 3 + 8 = 11 (over)
For x = 2:
3 + 2^(3(2)) = 3 + 2^6 = 3 + 64 = 67 (over)
Based on the over/under table, we can deduce that the solution to 3 + 2^(3x) = 9 is between x = 1 and x = 2.
Let's refine the estimate:
For x = 1.5:
3 + 2^(3(1.5)) = 3 + 2^4.5 ≈ 3 + 22.627 = 25.627 (over)
For x = 1.3:
3 + 2^(3(1.3)) = 3 + 2^3.9 ≈ 3 + 14.5 = 17.5 (over)
For x = 1.2:
3 + 2^(3(1.2)) = 3 + 2^3.6 ≈ 3 + 11.314 = 14.314 (over)
For x = 1.1:
3 + 2^(3(1.1)) = 3 + 2^3.3 ≈ 3 + 9.535 = 12.535 (over)
For x = 1.05:
3 + 2^(3(1.05)) = 3 + 2^3.15 ≈ 3 + 8.547 = 11.547 (over)
Based on the over/under table, we can estimate the solution to 3 + 2^(3x) = 9 to be between x = 1.05 and x = 1.1.
Therefore, the approximate solution to the equation is x ≈ 1.08 (rounded to two decimal places).