How do I solve for x in: 1/x > x
I need to do it both algebraically and graphically, but I'm not sure how to do so.
Thanks!
To solve the inequality 1/x > x algebraically, follow these steps:
1. Begin by multiplying both sides of the inequality by x to eliminate the fraction: x * (1/x) > x * x.
This simplifies to 1 > x^2.
2. Next, rearrange the inequality so that the quadratic term is on one side: x^2 - 1 < 0.
3. Factorise the inequality by expressing x^2 - 1 as (x - 1)(x + 1) < 0.
4. Set each factor to zero and determine the sign of the expression between the two critical points: x - 1 < 0 and x + 1 > 0.
Solving these inequalities, you find x < 1 and x > -1.
Therefore, the solution to 1/x > x is -1 < x < 0 or x > 1.
To solve the inequality graphically:
1. Graph the equation y = 1/x and the line y = x on the same coordinate system.
2. Identify the points where the two graphs intersect. These points represent the values of x where 1/x is equal to x.
3. Based on the inequality 1/x > x, you are looking for the regions where the graph of y = 1/x is located above the graph of y = x.
4. Shade the regions above the line y = x. The resulting shaded regions represent the values of x that satisfy the inequality.
Using this method, you will find the same solution as in the algebraic solution: -1 < x < 0 or x > 1.