The first inequality can be solved by factoring the quadratic expression and analyzing the sign of the factors:
x² - x + 6 > 0
The discriminant is negative (Δ = (-1)² - 4(1)(6) = -23), indicating that there are no real roots. Therefore, the quadratic expression is always positive, and the inequality holds for all values of x.
x² - x + 6 > 0 is true for all x.
The second inequality can be solved by applying the zero product property:
(х-5)(х+5) ≤ 0
The factors can have opposite signs or be equal to zero for this inequality to hold. Therefore, we can set up the following intervals:
x ≤ -5 or x ≥ 5
The solution to the inequality is the union of these two intervals:
Объяснение:
The first inequality can be solved by factoring the quadratic expression and analyzing the sign of the factors:
x² - x + 6 > 0
The discriminant is negative (Δ = (-1)² - 4(1)(6) = -23), indicating that there are no real roots. Therefore, the quadratic expression is always positive, and the inequality holds for all values of x.
x² - x + 6 > 0 is true for all x.
The second inequality can be solved by applying the zero product property:
(х-5)(х+5) ≤ 0
The factors can have opposite signs or be equal to zero for this inequality to hold. Therefore, we can set up the following intervals:
x ≤ -5 or x ≥ 5
The solution to the inequality is the union of these two intervals:
x ≤ -5 or x ≥ 5
{x | x ≤ -5 or x ≥ 5}