Question

How do I use a sign chart to solve `x^2<4`?

Answer 1

`x^2 < 4` when `-2 < x < 2`.

Explanation:

Let's modify the inequality by subtracting `4` from each side. Then

`x^2 – 4 < 0`

We start by finding the critical numbers.

Set `f(x) = x^2 – 4 = 0` and solve for `x`.

`(x+2)(x - 2) = 0`

`x+2 = 0` or `x-2 = 0`

`x = -2` or `x = +2`

The critical numbers are `-2` and `+2`.

Now we check for positive and negative regions.

We have three regions to consider: (a) `x < -2`; (b) `-2< x <2`; and (c) `x >2`.

Case (a): Let `x = -3`.

Then `f(-3) = (-3)^2 - 4 = 9-4 = 5`

`f(x) > 0` when `x < -2`.

Case (b): Let `x = 0`.

Then `f(0) = 0^2 -4 = 0-4 = -4`

`f(x) < 0` when `-2 < x < 2`

Case (c): Let `x = 3`.

Then `f(3) = 3^2 - 4 = 9-4 = 5`

`f(x) > 0` when `x > 2`.

If `x^2 -4 < 0` when `-2 < x < 2`, then

`x^2 < 4` when `-2 < x < 2`.