Solve log3(x 1)^2<log3(1-4x)

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

- 5 - 2 < x < 5 - 2

Alternative Form

x \in (- 5 - 2, 5 - 2)

Evaluate

lo g_{10} (3) \times (x \times 1)^{2} < lo g_{10} (3) \times (1 - 4 x)

Any expression multiplied by 1 remains the same

lo g_{10} (3) \times x^{2} < lo g_{10} (3) \times (1 - 4 x)

Move the expression to the left side

lo g_{10} (3) \times x^{2} - lo g_{10} (3) \times (1 - 4 x) < 0

Subtract the terms

More Steps

lo g_{10} (3) \times x^{2} - lo g_{10} (3) + 4 lo g_{10} (3) \times x < 0

Rewrite the expression

lo g_{10} (3) \times x^{2} - lo g_{10} (3) + 4 lo g_{10} (3) \times x = 0

Add or subtract both sides

lo g_{10} (3) \times x^{2} + 4 lo g_{10} (3) \times x = lo g_{10} (3)

Divide both sides

\frac{lo g _{10} ( 3 ) \times x ^{2} + 4 lo g _{10} ( 3 ) \times x}{lo g _{10} ( 3 )} = \frac{lo g _{10} ( 3 )}{lo g _{10} ( 3 )}

Evaluate

x^{2} + 4 x = 1

Add the same value to both sides

x^{2} + 4 x + 4 = 1 + 4

Simplify the expression

(x + 2)^{2} = 5

Take the root of both sides of the equation and remember to use both positive and negative roots

x + 2 = \pm 5

Separate the equation into 2 possible cases

x + 2 = 5 x + 2 = - 5

Move the constant to the right-hand side and change its sign

x = 5 - 2 x + 2 = - 5

Move the constant to the right-hand side and change its sign

x = 5 - 2 x = - 5 - 2

Determine the test intervals using the critical values

x < - 5 - 2 - 5 - 2 < x < 5 - 2 x > 5 - 2

Choose a value form each interval

x_{1} = - 5 x_{2} = - 2 x_{3} = 1

To determine if x < - 5 - 2 is the solution to the inequality,test if the chosen value x = - 5 satisfies the initial inequality

More Steps

x < - 5 - 2 is not a solution x_{2} = - 2 x_{3} = 1

To determine if - 5 - 2 < x < 5 - 2 is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < - 5 - 2 is not a solution - 5 - 2 < x < 5 - 2 is the solution x_{3} = 1

To determine if x > 5 - 2 is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < - 5 - 2 is not a solution - 5 - 2 < x < 5 - 2 is the solution x > 5 - 2 is not a solution

Solution

- 5 - 2 < x < 5 - 2

Alternative Form

x \in (- 5 - 2, 5 - 2)

Show Solution