Question

1) (х+2)(х-3)<0 2) x-1/x+1≤0

Answer 1

1)(x+2)(x-3)<0
x²-3x+2x-6<0
x²-x-6<0
x<-2;3
2)x-1/x+1≤0
x-1≤0
x≤1

Answer 2

To solve the inequalities, we need to find the values of x that satisfy the given conditions. Let's solve them step by step:

1. (x + 2)(x - 3) < 0

To solve this inequality, we can use the concept of intervals. First, let's find the critical points where the expression becomes zero (i.e., where the factors are equal to zero):

x + 2 = 0 => x = -2 x - 3 = 0 => x = 3

Now, we have three intervals to check: (-∞, -2), (-2, 3), and (3, ∞). We can pick a test point from each interval and evaluate the expression (x + 2)(x - 3) to determine the sign of the expression in each interval.

Test points:

• Let's pick x = -3 (from (-∞, -2)): (-3 + 2)(-3 - 3) = (-1)(-6) = 6 (positive)

• Let's pick x = 0 (from (-2, 3)): (0 + 2)(0 - 3) = (2)(-3) = -6 (negative)

• Let's pick x = 4 (from (3, ∞)): (4 + 2)(4 - 3) = (6)(1) = 6 (positive)

The signs change from positive to negative and then back to positive. The inequality (x + 2)(x - 3) < 0 is satisfied when the expression is negative. Therefore, the solution to this inequality is:

x ∈ (-2, 3)

2. x - 1 / x + 1 ≤ 0

To solve this inequality, first, let's simplify the expression:

x - 1 / x + 1 ≤ 0 (x^2 - 1) / (x + 1) ≤ 0 ((x + 1)(x - 1)) / (x + 1) ≤ 0

Note that we can cancel out the common factor (x + 1) since x ≠ -1:

x - 1 ≤ 0

Now, let's solve this simple inequality:

x ≤ 1

The solution to the inequality x - 1 / x + 1 ≤ 0 is:

x ≤ 1