Approximate solving and rational inequalities
Answers are for parents and teachers.
1
Find an interval with a root of f(x) = xΒ³ + xΒ² β 5 and narrow it by one bisection step.
f(1) = β3 < 0, f(2) = 7 > 0 β (1, 2). f(1.5) = 0.625 > 0 β the root is in (1, 1.5).
2
Find integer-ended intervals for the three roots of xΒ³ β 6x + 3 = 0.
f(β3) = β6, f(β2) = 7 β (β3, β2); f(0) = 3, f(1) = β2 β (0, 1); f(2) = β1, f(3) = 12 β (2, 3).
3
Solve (x + 3)(x β 2)(5 β x) β€ 0.
(x + 3)(x β 2)(x β 5) β₯ 0 β [β3, 2] βͺ [5, β).
4
Solve (xΒ² β 9)/(x β 2)Β² > 0.
x β 2, (x β 3)(x + 3) > 0 β (ββ, β3) βͺ (3, β).
5
Solve the system: xΒ² β 4x + 3 < 0 and (x β 2)/(x + 5) β₯ 0.
The first gives (1, 3), the second (ββ, β5) βͺ [2, β); the common part is [2, 3).