Explanation

The magnitude (length) of a vector is always ≥ 0.

It can be zero (for a zero vector) or positive, but never negative.

Explanation

To solve the inequality 4-x>5, you need to isolate the variable x.

Subtract 4 from both sides of the inequality: 4-x-4>5-4

-x>1

Multiply both sides by -1.

When you multiply or divide an inequality by a negative number, you must reverse the inequality sign.

(-1)(-x)<(-1)(1)

x<-1

Explanation

The domain of f(x)= log x is the set of all positive real numbers, which can be expressed in interval notation as 0, ∞.

Explanation

Degrees to Radians

Radians = θ° × π/180

For 120°: 120 × π/180 = 120/180π = 2π/3

Explanation

Quadratic Equation

Given: x^2 + x - 6 = 0

Let's factor it:

(x + 3)(x - 2) = 0

x + 3 = 0 or x - 2 = 0

x = -3 or x = 2

Explanation

Quadratic Equation Roots

Given: ax^2 + bx + c = 0 with a > 0$ and roots greater than 1.

For both roots to be greater than 1, given the parabola opens upwards (a > 0), the condition is f(1) > 0:

f(1) = a(1)^2 + b(1) + c = a + b + c > 0.

Explanation

The sine function sinθ is defined for all real values of θ.

Its range is bounded (−1 ≤ sinθ ≤ 1), but the domain covers all reals.

Explanation

Trigonometric Identity

Given: sin a

The double-angle formula for sine is sin 2a = 2 sin a cos a, but there's also a relation involving half-angles: sin a = 2 sin a/2 cos a/2.

Explanation

√23 cannot be expressed as a fraction

𝑝/𝑞 since 23 is not a perfect square.

Its decimal expansion is non-terminating and non-repeating, so it is irrational.

Explanation

Trigonometric Identity

Given: cos^2 θ sec^2 θ

Since sec θ = 1/cosθ,

cos^2 θ sec^2 θ = cos^2 θ . 1/cos^2 θ = 1.