Introduction to inequalities

Correct answer: C. ( \emptyset )

For every real (x), (x^2\geq 0), so (x^2<0) is impossible. In such questions, check the basic property first.

Correct answer: C. (x\geq 14)

Since (3) is positive, the sign does not change and (x-2\geq 12) gives (x\geq 14). With positive multiplication, the sign remains the same.

Correct answer: B. (x+1>-8) so (x>-9)

Multiplying by ( -4 ) reverses the sign to (x+1>-8). Be extra careful when the denominator is negative.

Correct answer: C. (x=0)

(3x-7\leq 2) gives (x\leq 3), and (x=0) satisfies it but not (x<0). Direct option checking is a fast method.

Correct answer: A. (p+r>q+s)

Adding inequalities in the same direction gives (p+r>q+s), which is always true. Subtraction and multiplication need extra conditions.

Correct answer: A. ( [-2,\infty) )

(5-2x\leq x+11) gives (-3x\leq 6), so (x\geq -2). Division by a negative coefficient reverses the sign.

Correct answer: B. (-5<x<5)

( |x|<a ) means (-a<x<a) when (a>0). For less-than distance questions, the solution is the middle interval.

Correct answer: A. (x\leq 1) or (x\geq 5)

( |x-3|\geq 2 ) means (x-3\leq -2) or (x-3\geq 2). Greater-distance questions give two outer intervals.

Correct answer: A. (x>-22)

Multiplying by positive (10) gives (4x-2<5x+20), so (x>-22). With a positive common denominator, the sign does not change.

Correct answer: B. (5)

(2x<11) gives (x<\frac{11}{2}), so the greatest natural number is (x=5). For integer domains, do not write a decimal answer.

Correct answer: B. Open dot at (4) and shading left

In ( (-\infty,4) ), (4) is not included and all smaller values are included. An open dot means the boundary is excluded.

Correct answer: C. Multiplying both sides by (-5)

Multiplication or division by a negative number reverses the inequality sign. Addition and subtraction do not change the sign.

Correct answer: A. (3\leq x<7)

Subtracting (1) throughout gives (6\leq 2x<14), then dividing by (2) gives (3\leq x<7). Work on both bounds together.

Correct answer: C. (7)

(-4\leq 2x<10) gives (-2\leq x<5), so (-2,-1,0,1,2,3,4) are (7) solutions. Check endpoints separately.

Correct answer: B. (x<1)

Since (m) is negative, division by (m) reverses the sign and gives (x<1). In parameter questions, check the sign of the coefficient first.

Correct answer: B. Not true for any (x\in \mathbb{R})

Both sides are the same (x+3), so the statement becomes (x+3>x+3), which is false. Recognize equal sides quickly.

Correct answer: C. All (x\in \mathbb{R})

Since (x^2\geq 0), (x^2+1\geq 1>0) for all real (x). Recognizing always positive forms is useful.

Correct answer: A. (x<6) and (x>-3)

The first inequality gives (x<6), and the second gives (x>-3), so (-3<x<6). In a chained inequality, both conditions must hold together.

Correct answer: B. (m>40)

More than (40) means (m>40), so (40) is not included. In word problems, identify whether the boundary is included.

Correct answer: B. (p\leq 500)

Not more than means at most (500), so (p\leq 500). Words like at most and at least are important.

Correct answer: A. (x+y>3)

Direct answer: option A, x+y>3. We are told x>1 and y>2. Add the inequalities term by term: x+y>1+2, so x+y>3. This conclusion is always true, although the sum need not equal 3; for example, x=2 and y=3 gives 5. Option A is correct because both quantities are strictly greater than their lower bounds. Option B, x+y<3, is wrong because choosing x=2,y=3 gives a sum of 5. Option C, x+y=3, is wrong because the values can be larger than 1 and 2, and strict inequalities prevent equality with the lower-bound sum. Option D, x+y≤3, is also wrong for the same example and is weaker in the wrong direction. Adding inequalities with the same direction is valid here. Memory cue: greater than plus greater than gives greater than the sum of the bounds.

Correct answer: B. (x=-2) is possible

The direct answer is B: x=−2 is possible. The condition −2<x≤4 has two parts: x must be strictly greater than −2, and x may be equal to 4. Thus −2 is excluded, while 4 is included. A is true because 4≤4. B is false because −2 is not greater than −2; equality is forbidden by the strict < sign. C is true because −2<0≤4. D is true because −2<3.5≤4. On a number line, place an open circle at −2 and a closed circle at 4, shading only the interval between them. The open endpoint represents exclusion, and the closed endpoint represents inclusion. Therefore the statement asking whether x=−2 is possible is the false one. A common mistake is to treat < and ≤ alike; the small extra line in ≤ includes the endpoint. Memory cue: open circle means not included, closed circle means included.

Correct answer: B. (x^2+4\leq 0)

(x^2+4) is always at least (4), so (x^2+4\leq 0) is impossible. Use the minimum value to judge solutions.

Correct answer: C. (a>c)

The same-direction chain gives (a>b>c), hence (a>c). This is the transitive property of inequalities.

Correct answer: A. ( -5\leq x<7 )

Multiplying by positive (2) gives (-2\leq x+3<10), then subtracting (3) gives (-5\leq x<7). Positive multiplication does not change direction.