Introduction to inequalities

Correct answer: B. (2-5x\geq 17)

Multiplying (x\leq -3) by (-5) gives (-5x\geq 15), then adding (2) gives (2-5x\geq 17). Negative multiplication changes direction.

Correct answer: D. (3n\geq 18)

Not less than means greater than or equal to, so (3n\geq 18). Pay attention to negative phrases while translating words into symbols.

Correct answer: C. True for all real (x)

The square of any real number is never negative, so ( (x-2)^2\geq 0 ) is always true. For squared forms, check the minimum value.

Correct answer: C. (x>1)

In (\mathbb{R}), outside (x\leq 1) lies only (x>1). In the complement, equality at the boundary is excluded.

Correct answer: A. (x\geq -11)

Multiplying by positive (4) gives (2x-10\leq 3x+1), so (x\geq -11). Track the sign carefully while moving variables.

Correct answer: A. ( -13<3x-1\leq 5 )

Multiplying the whole inequality by (3) and subtracting (1) gives ( -13<3x-1\leq 5 ). Positive multiplication does not change the sign.

Correct answer: A. (5)

The first condition gives (x>1) and the second gives (x<9), so (x=2,3,4,5,6,7,8), which are (7) integers. Always list integer endpoints carefully.

Correct answer: A. (0<\frac{x}{-2}\leq 3)

Dividing by (-2) reverses the direction and gives (3\geq \frac{x}{-2}>0). In increasing order, it is (0<\frac{x}{-2}\leq 3).

Correct answer: B. (2x-5\geq 3)

(2x-5\geq 3) gives (x\geq 4), so (4) is included and (3) is not. Always check equality at boundary options.

Correct answer: B. (ab<0)

The product of a negative and a positive number is always negative, so (ab<0). In sign-based questions, also check the possibility of zero.

Correct answer: C. There is no real solution

The square of any real number cannot be less than (0). Therefore ( (x+1)^2<0 ) has no real solution.

Correct answer: A. (12<t+5<20)

Both greater than and less than are strict phrases, so (12<t+5<20) is correct. In word problems, first decide whether equality is included.