how do i solve this equation: 3x^2-1 is greater than or equal to (1/2) (3+x^2)the (1/2) in parentheses is a fraction

embizze | High School Teacher | (Level 1) Educator Emeritus

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Solve `3x^2-1 >= (3+x^2)/2` .

First we rewrite as an equivalent inequality (one that has the same solutions as the original)

`3x^2-1 >= (3+x^2)/2`

`6x^2-2>=3+x^2` Multiply both sides by 2

`5x^2-5>=0` Pull terms to left side by subtraction

`5(x^2-1)>=0 => 5(x+1)(x-1)>=0` Factor.

Now the left hand side is zero when x=1 or -1. To find the intervals where it is greater than zero we plug in test values. The intervals we are interested in are x<-1, -1<x<1, and x>1; so we try x=-2,x=0, and x=2.

If x=-2 the left hand side is positive; so if x<-1 then the left-hand side is greater than zero.

If x=0 the left-hand side is negative so the left-hand side is negative for -1<x<1.

If x=2 the LHS is positive so LHS is greater than zero if x>1.

We can also look at the graph to check our work -- notice that the "skinnier" parabola is above the wider parabola as long as x<-1 or x>1.

Thus the solution to `3x^2-1 >= (3+x^2)/2` is `x<= -1` or `x>=1` .