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

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

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

𝜋/3 is irrational, and dividing it by 3 keeps it irrational.

Hence, 𝜋/3 cannot be written as a fraction of integers.

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.

Explanation

In algebra, rational fractions are classified into 3 types:

1. Proper fractions → numerator degree < denominator degree.
2. Improper fractions → numerator degree ≥ denominator degree.
3. Mixed fractions → combination of a polynomial and a proper fraction.

Explanation

Number of Solutions

Given: √y+ √y + 3 = 1

Let's solve it:

√y + 3 = 1 - √y

Squaring both sides:

y + 3 = 1 - 2 √y + y

2 = -2 √y

√y = -1

Since √y = 0 for all y = 0, there's no real solution for y.

Explanation

This is from system of linear equations (AX = 0).

If |A| ≠ 0, the system has only the trivial solution (x = 0).

If ∣A∣ = 0, then non-trivial solutions may exist.