Find the range of values of x for which \(2x^{2} + 7x - 15 >...

Find the range of values of x for which \(2x^{2} + 7x - 15 > 0\).

\(2x^{2} + 7x - 15 > 0 \implies 2x^{2} - 3x + 10x - 15 > 0\)

\(x(2x - 3) + 5(2x - 3) > 0\)

\((x + 5)(2x - 3) > 0\)

For their product to be positive, they are either both +ve or -ve.

\(x + 5 > 0 \implies x > -5\)

\(2x - 3 > 0 \implies 2x > 3\)

\(x > \frac{3}{2}\)

Check:

\(x > -5: x = -3\)

\(2(-3)^{2} + 7(-3) - 15 = 18 - 21 - 15 = -18 < 0\) (Not satisfied)

\(\therefore x < -5\)

\(x > \frac{3}{2}: x = 2\)

\(2(2^{2}) + 7(2) - 15 = 8 + 14 - 15 = 7 > 0\) (Satisfied)

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