Radical Inequalities and Their Solutions

Miss Jones introduces radical inequalities, explaining their similarities to radical equations and the importance of ensuring non-negative values under square roots. She provides multiple examples, demonstrating how to solve these inequalities by isolating the radical, converting inequalities to equations, and testing values. The video concludes with instructions for further practice and availability for questions.

What is a key restriction when dealing with square roots in radical inequalities?

• The radicand must be zero.

• The radicand must be positive.

• The radicand can be any real number.

• The radicand must be negative.

In the first example, what is the solution set for the inequality \( \sqrt{3x + 3} \leq 6 \)?

• [-1, ∞)

• (-∞, -1]

• [-1, 11]

• [0, 11]

What is the first step in solving a radical inequality with a non-isolated radical?

• Isolate the radical.

• Set the inequality to zero.

• Add a constant to both sides.

• Multiply both sides by a constant.

In the second example, what is the solution set for the inequality \( -2\sqrt{x + 1} < -8 \)?

• [15, ∞)

• (-∞, 15)

• (-∞, 15]

• (15, ∞)

What is the solution set for the inequality \( \sqrt{2x + 1} \geq -4 \)?

• [-1/2, ∞)

• (-∞, 7.5]

• [7.5, ∞)

• (-∞, -1/2]

In the third example, what is the solution set for the inequality \( 9 \geq 5 + \sqrt{x - 2} \)?

• [2, 18)

• (2, 18]

• [2, 18]

• (2, 18)

How does the handling of cube roots differ from square roots in radical inequalities?

• Cube roots must have zero as the radicand.

• Cube roots must have negative radicands.

• Cube roots can have any real number as the radicand.

• Cube roots can only have positive radicands.

In the final example, what is the solution set for the inequality \( \sqrt[3]{4x - 1} \leq 3 \)?

• [7, ∞)

• (-∞, 7]

• (-∞, 7)

• [7, ∞)

What is the result of cubing both sides of the equation \( \sqrt[3]{4x - 1} = 3 \)?

• 4x - 1 = 3

• 4x - 1 = 27

• 4x - 1 = 81

• 4x - 1 = 9

What is the correct interval notation for the solution set of \( \sqrt[3]{4x - 1} \leq 3 \)?

• [7, ∞)

• (-∞, 7)

• [7, ∞)

• (-∞, 7]