How to find an interval that contains a solution?

Question

These are similar to some problems I was assigned for homework ad I tried googling them, but I don't think I even understand what it is asking me

For each of the following equations, find an interval that contains a solution

a) x^7 + 3x = 2

b) x^4 + x = 3

c) log3 (x^2 ) + √ x = 0

Answer 1

For equation a), if you create the function f(x) = x⁷+3x - 2, and consider where the function might change from positive to negative (or vice versa), you can narrow in an on interval that contains a solution, i.e. when does f(x) = 0?

For example f(0) = -2 and f(1) = 2. Since f is continuous (everywhere), the function must cross the x-axis somewhere between 0 and 1. So the interval (0,1) contains a root for f. Graph f on a graphing calculator (Desmos) to see that in fact there is a root that is approx .65.

Follow a similar pattern for parts b and c.