Question

Solve 2x² + x – 15 ≤ 0

Answer 1

The given inequality is

2x² + x – 15 ≤ 0   ......(1)

2x² + x – 15 = 2x² + 6x – 5x – 15

= 2x(x + 3) – 5(x + 3)

= (2x – 5)(x + 3)

2x² + x – 15 = `2(x - 5/2)(x + 3)`  ......(2)

The critical numbers are `x - 5/2` = 0 or x + 3 = 0

The critical numbers are x = `5/2` or x = – 3

Divide the number line into three intervals

`(- oo, - 3), (- 3, 5/2)` and `(5/2, oo)`

(i) `(– oo, – 3)`

When x < – 3 say x = – 4

The factor `x - 5/2 = -  4 - 5/2 < 0` and

x + 3 = – 4 + 3 = – 1 < 0

`x - 5/2 < 0` and x + 3 < 0

⇒ `(x - 5/2) (x + 3) > 0`

Using equation (2) 2x² + x – 15 > 0

∴ 2x² + x – 15 ≤ 0 is not true in (– ∞, – 3)

(ii) `(- 3, 5/2)`

When `- 3 < x < 5/2` say x = 0

The factor `x - 5/2 = 0 - 5/2 = - 5/2 < 0` and
x + 3 = 0 + 3 = 3 > 0

`x - 5/2 < 0` and x + 3 > 0

⇒ `(x - 5/2) (x + 3) < 0`

Using equation (2) 2x² + x – 15 < 0

∴ 2x² + x – 15 ≤ 0 is true in `(- 3, 5/2)`

(iii) `(5/2, oo)`

When `x > 5/2` say x = 3

The factor `x - 5/2 = 3 - 5/2 > 0` and

x + 3 = 3 + 3 > 0

`x - 5/2 > 0` and x + 3 > 0

⇒ `(x - 5/2) (x + 3) > 0`

Using equation (2) 2x² + x – 15 > 0

∴ 2x² + x – 15 ≤ 0 is true in `(5/2, ∞)`

Internal	Sign of `x - 5/2`	Sign of x + 3	Sign of 2x2 + x – 15
`(– oo, – 3)`	–	–	+
`(- 3, 5/2)`	–	+	–
`(5/2,  oo)`	+	+	+

We have proved the inequality 2x² + x – 15 ≤ 0 is true in the interval

But it is not true in the interval

`(– oo, – 3)` and `(5/2,  oo)`

∴ The solution set is`[- 3, 5/2]`