What is |2x-1|<sqrt(x^2-x+1/(|-4|))+1 ?

The solution to |2x-1|<sqrt(x^2-x+1/(|-4|))+1 is -1/2 <x< 3/2

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|2x-1|<sqrt(x^2-x+1/(|-4|))+1

∣ 2 x - 1 ∣ < x^{2} - x + \frac{1}{∣ - 4 ∣} + 1

- \frac{1}{2} < x < \frac{3}{2}

Solution steps

∣ 2 x - 1 ∣ < x^{2} - x + \frac{1}{∣ - 4 ∣} + 1

Switch sides

x^{2} - x + \frac{1}{∣ - 4 ∣} + 1 > ∣ 2 x - 1 ∣

Subtract

1

from both sides

x^{2} - x + \frac{1}{∣ - 4 ∣} + 1 - 1 > ∣ 2 x - 1 ∣ - 1

Simplify

x^{2} - x + \frac{1}{∣ - 4 ∣} > ∣ 2 x - 1 ∣ - 1

f (x) > g (x)

then

g (x) < 0 or g (x) \geq 0

For

∣ 2 x - 1 ∣ - 1 < 0 : 0 < x < 1

For

∣ 2 x - 1 ∣ - 1 \geq 0 : - \frac{1}{2} < x \leq 0 or 1 \leq x < \frac{3}{2}

Combine the intervals

- \frac{1}{2} < x \leq 0 or 1 \leq x < \frac{3}{2} or 0 < x < 1

Merge Overlapping Intervals

- \frac{1}{2} < x < \frac{3}{2}

Verify Solutions:

- \frac{1}{2} < x < \frac{3}{2}

True

The solution is

- \frac{1}{2} < x < \frac{3}{2}

x^{2} + 3 x - 54

9 k^{2}

0 = - x^{2} + 44 x + 1

\frac{k}{6.5} = - 15.7

8 c^{2} + 11 c + 7 = 6 c^{2}

What is |2x-1|<sqrt(x^2-x+1/(|-4|))+1 ?
The solution to |2x-1|<sqrt(x^2-x+1/(|-4|))+1 is -1/2 <x< 3/2