Question

How do you solve `-2 + 7x - 3x^2<0`?

Answer 1

(`-∞, 1/3`) and (`2, ∞`)

Explanation:

`-2 + 7x -3x^2 < 0`

One way to solve this inequality is to use a sign chart.

Step 1. Write the inequality in standard form.

`-3x^2 +7x -2 < 0`

Step 2. Multiply the inequality by `-1`.

`3x^2 -7x +2 > 0`

Step 3. Find the critical values.

Set `f(x) =3x^2 -7x +2 = 0` and solve for `x`.

`(3x-1)(x-2) = 0`

`3x-1 = 0` or `x-2 = 0`

`x = 1/3` or `x = 2`

The critical values are `x = 1/3` and `x = 2`.

Step 4. Identify the intervals.

The intervals to consider: (`-∞, +1/3`), (`1/3, 2`), and (`2, ∞`).

Step 5. Evaluate the intervals.

We pick a test number and evaluate the function and its sign at that number.

Step 6. Create a sign chart for the function.

The only intervals for which the signs are positive are.

Solution: `x < 1/3`or `x > 2`