Solve (4x - 5)/3 > sqrt(3 - x)

Question

Solve the inequality

2 < x \leq 3

Alternative Form

x \in (2, 3]

Evaluate

\frac{4 x - 5}{3} > 3 - x

Find the domain

More Steps

\frac{4 x - 5}{3} > 3 - x, x \leq 3

Multiply both sides of the inequality by 3

\frac{4 x - 5}{3} \times 3 > 3 - x \times 3

Multiply the terms

4 x - 5 > 3 - x \times 3

Multiply the terms

4 x - 5 > 3 3 - x

Calculate

4 x - 5 = 3 3 - x

Move the expression to the left side

4 x - 5 - 3 3 - x > 0

Move the expression to the right side

- 3 3 - x > - 4 x + 5

Change the signs on both sides of the inequality and flip the inequality sign

3 3 - x < 4 x - 5

Separate the inequality into 2 possible cases

3 3 - x < 4 x - 5, 4 x - 5 \geq 0 3 3 - x < 4 x - 5, 4 x - 5 < 0

Solve the inequality

More Steps

x \in (- \infty, - \frac{1}{16}) \cup (2, + \infty), 4 x - 5 \geq 0 3 3 - x < 4 x - 5, 4 x - 5 < 0

Solve the inequality

More Steps

x \in (- \infty, - \frac{1}{16}) \cup (2, + \infty), x \geq \frac{5}{4} 3 3 - x < 4 x - 5, 4 x - 5 < 0

Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is false for any value of x

x \in (- \infty, - \frac{1}{16}) \cup (2, + \infty), x \geq \frac{5}{4} x \in \emptyset, 4 x - 5 < 0

Solve the inequality

More Steps

x \in (- \infty, - \frac{1}{16}) \cup (2, + \infty), x \geq \frac{5}{4} x \in \emptyset, x < \frac{5}{4}

Find the intersection

x > 2 x \in \emptyset, x < \frac{5}{4}

Find the intersection

x > 2 x \in \emptyset

Find the union

x > 2

Check if the solution is in the defined range

x > 2, x \leq 3

Solution

2 < x \leq 3

Alternative Form

x \in (2, 3]

Show Solution