Solving Inequalities with Exponential Functions

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In summary, the conversation discusses solving the inequality x + 3^x < 4 by using the sum of an identity function and an exponential function. It is determined that the function is monotone increasing and crosses y=4 at only one x-value, x_1. The solution to the inequality is x < x_1 and can be found by solving for x in x + 3^x = 4. Newton's method is suggested for finding an approximate solution and it is noted that in general, equations such as a^x+x=b cannot be solved in a closed form. However, in this case, the exact solution is x=1.
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verty
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I'm wondering how to solve an inequality like this:

x + 3^x < 4

I can see that it is the sum of the identity function and an exponential function. I can see that it is monotone increasing because each of those is. I therefore know that it crosses y=4 at only one x-value (call it x_1) and I know the solution will be x < x_1.

So I need to solve for x in x + 3^x = 4, how would I do that? I see I could draw the graph and read off the value, which I can see is x_1 = 1, but is there a way to calculate it?

Thank you for any clarification.
 
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  • #2
verty said:
x + 3^x < 4

Try this: [itex]x+3^x=4+k[/itex] for [itex]k>0[/itex].
 
  • #3
Draw a graph first, let the function be y = x + 3^x - 4 and find where this graph cuts the x-axis (That will satisfy the Ineq, Find an approximate area where the graph cuts the axis, so accuracy is paramount.). From here you should have an idea of the kind of answer you'll need. To find a approximate solution, I'd use Newton's method which is:
x_n+1 = x_n - F(x_n)/F'(x_n)
You can take the first value of x_n from the graph you've drawn.
 
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  • #4
Thank you for suggesting Newton's method.
 
  • #5
In general, you can't solve equations like a^x+x=b in a closed form, although you can approximate the solution to any degree of accuracy. However in this case, it's easy to check that the solution is exactly x=1.
 

What is the inequality x + 3^x < 4 asking us to solve?

The inequality x + 3^x < 4 is asking us to find all possible values of x that will make the expression less than 4.

What are the steps to solving this inequality?

To solve this inequality, we must first rearrange the equation to isolate the variable by subtracting 3^x from both sides. This will give us x < 4 - 3^x. Then, we can use a graphing calculator or a table of values to determine the range of x values that satisfy this inequality.

What is the range of values for x that satisfy the inequality?

The range of values for x that satisfy the inequality x + 3^x < 4 is x < 1. This means that any value of x less than 1 will make the expression less than 4.

Is there more than one solution to this inequality?

Yes, there are infinitely many solutions to this inequality. This is because any value of x that is less than 1 will satisfy the inequality, including negative numbers and decimal numbers.

Can this inequality be solved algebraically?

No, this inequality cannot be solved algebraically because the variable is present in both the base and exponent of the expression. We must use a graphing calculator or a table of values to determine the range of x values that satisfy the inequality.

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