Solve for x 2/3x+8/3=2^x

Problem

2/3*x+8/3=2^x

Solution

1. Identify the type of equation. This is a transcendental equation involving both a linear term and an exponential term, which often requires inspection or numerical methods.

2. Test small integer values for x to see if they satisfy the equation.

3. Substitute x=2 into the left side:

2/3*(2)+8/3=4/3+8/3=12/3=4

4. Substitute x=2 into the right side:

2^2=4

5. Verify another potential solution by testing x=−4 Substitute x=−4 into the left side:

2/3*(−4)+8/3=−8/3+8/3=0

6. Substitute x=−4 into the right side:

2^(−4)=1/16

7. Observe that 0≠1/16 so x=−4 is not a solution.

8. Test x=4 into the left side:

2/3*(4)+8/3=8/3+8/3=16/3

9. Substitute x=4 into the right side:

2^4=16

10. Observe that 16/3≠16

11. Analyze the functions ƒ(x)=2/3*x+8/3 and g(x)=2^x graphically. The linear function and the exponential function intersect at exactly two points.

12. Test x=−3 into the left side:

2/3*(−3)+8/3=−2+8/3=2/3

13. Substitute x=−3 into the right side:

2^(−3)=1/8

14. Test x=−2 into the left side:

2/3*(−2)+8/3=−4/3+8/3=4/3

15. Substitute x=−2 into the right side:

2^(−2)=1/4

16. Test x=3 into the left side:

2/3*(3)+8/3=2+8/3=14/3≈4.67

17. Substitute x=3 into the right side:

2^3=8

18. Conclude that x=2 is the only integer solution. Numerical analysis shows a second irrational solution near x≈−3.88

Final Answer

x=2

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