Solve (2m-1)^2-4(m^2-1)>0

$$( 2 m - 1 ) ^ { 2 } - 4 ( m ^ { 2 } - 1 ) > 0$$

Solve for m

$$m<\frac{5}{4}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2m-1\right)^{2}.

$$4m^{2}-4m+1-4\left(m^{2}-1\right)>0$$

Use the distributive property to multiply -4 by m^{2}-1.

$$4m^{2}-4m+1-4m^{2}+4>0$$

Combine 4m^{2} and -4m^{2} to get 0.

$$-4m+1+4>0$$

Add 1 and 4 to get 5.

$$-4m+5>0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$-4m>-5$$

Divide both sides by -4. Since -4 is negative, the inequality direction is changed.

$$m<\frac{-5}{-4}$$

Fraction \frac{-5}{-4} can be simplified to \frac{5}{4} by removing the negative sign from both the numerator and the denominator.

$$m<\frac{5}{4}$$