Introduction to inequalities

Correct answer: A. (x<-3)

Dividing by a negative number reverses the inequality sign. In exams, always reverse the sign when dividing by (-3).

Correct answer: B. (7)

The inequality gives (2x\leq 14), so (x\leq 7). With an integer restriction, choose an integer final answer.

Correct answer: A. (x\geq 11)

Simplifying gives (3x-6\geq 2x+5), so (x\geq 11). While opening brackets, handle signs and constants carefully.

Correct answer: A. (x\leq 7)

Multiplying by positive (3) does not change the sign and gives (x-1\leq 6). If the denominator is positive, the direction stays the same.

Correct answer: A. ({1,2,3,4,5})

The inequality gives (3x<18), so (x<6). In (\mathbb{N}), usually take positive integers starting from (1).

Correct answer: A. ({-1,0,1,2,3})

From (-1<2x+3\leq 9), we get (-4<2x\leq 6), hence (-2<x\leq 3). For integers, handle open and closed endpoints carefully.

Correct answer: A. (x\geq 3)

From (7-3x\leq -2), (-3x\leq -9), so (x\geq 3). Dividing by a negative coefficient reverses the sign.

Correct answer: A. (c\in\mathbb{R})

Adding the same real number does not change the direction of inequality. For addition, the sign of (c) does not matter.

Correct answer: A. ([2,5))

From (0\leq 4x-8<12), we get (8\leq 4x<20), hence (2\leq x<5). Solve both ends of a compound inequality together.

Correct answer: A. (x\leq 3)

Multiplying by (10) gives (2x+4\geq 5x-5), so (9\geq 3x). Multiplication by a positive LCM does not change the sign.

Correct answer: A. ([-2,3))

Taking both conditions together gives (-2\leq x<3). The combined solution is the intersection of both inequalities.

Correct answer: A. (x\geq 1)

From (9\leq 3x+6), we get (3\leq 3x), hence (x\geq 1). Read the meaning carefully when the inequality is written in reverse order.

Correct answer: A. (5\geq x)

The direct answer is A: 5≥x. The statement x≤5 says that x is less than or equal to 5. Reversing the order of the two sides gives the equivalent statement 5≥x; the meaning does not change. For example, x=5 satisfies both, and x=2 also satisfies both. A is correct because “x is at most 5” means “5 is at least x.” B, 5<x, means x<5 and excludes x=5, so it is not equivalent. C, x>5, describes values greater than 5, the opposite direction. D, 5≤x, means x≥5 and includes values greater than or equal to 5, also the reverse inequality. Note that merely changing sides changes the visible direction of the sign, but it does not mean multiplying by a negative number. The boundary value 5 must remain included because the original sign is ≤. Memory cue: read ≤ as “at most”; reversing sides changes it to ≥.

Correct answer: A. (3) पर खुला वृत्त और बाईं ओर छाया

In (x<3), (3) is not included and smaller numbers are included. Use an open circle for a strict inequality.

Correct answer: A. (x\geq -5)

From (3x+4\geq x-6), (2x\geq -10), so (x\geq -5). Keeping like terms organized reduces mistakes.

Correct answer: A. (p-7>q-7)

Subtracting the same number from both sides does not change the direction. Squares and reciprocals need extra conditions.

Correct answer: A. (5)

The direct answer is A: 5. Start with 4x−1>15. Add 1 to both sides: 4x>16. Divide by the positive number 4: x>4. Because x must be an integer, the smallest integer greater than 4 is 5. A is correct, and checking it gives 4(5)−1=19>15. B, 4, is not allowed because 4(4)−1=15, and 15 is not greater than 15; the inequality is strict. C, 6, satisfies the inequality but is not the least possible value. D, 3, gives 11>15, which is false. Dividing by a positive number keeps the sign unchanged; if we divided by a negative number, the sign would reverse, but that does not happen here. The integer condition is essential: among real numbers there is no least real number greater than 4, but among integers the next value is 5. Memory cue: for x>4 with x∈Z, move to the next integer, not 4.

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

Direct answer: option A, 3≤x<7. Solve each inequality separately. From 2x+1≥7, subtract 1 on both sides: 2x≥6. Divide by the positive number 2: x≥3; the inequality sign stays the same. From x-4<3, add 4: x<7. The word “and” means both conditions must hold at once, so take their overlap: x is at least 3 but less than 7, written 3≤x<7. Option A is correct. Option B reverses which endpoint is included and excluded. Option C keeps only x≥7, which does not follow and misses values such as 4. Option D keeps only x<3, which contradicts the first inequality and misses valid values such as 3. On a number line, use a closed dot at 3 and an open dot at 7.

Correct answer: A. \(2\leq x<6\)

Option A, \(2\leq x<6\), includes \(x=2\) because the symbol next to 2 is \(\leq\), but excludes \(x=6\) because the symbol next to 6 is strict \(<\). Option D also includes 6, while options B and C exclude 2. Exam tip: \(\leq\) and \(\geq\) include an endpoint, whereas \(<\) and \(>\) exclude it.

Correct answer: A. (a>-3)

Dividing by (-2) reverses the inequality to (a>-3). Division by a negative number is the most common mistake point.

Correct answer: A. (x>1)

From (6<2x+4), we get (2<2x), hence (x>1). Even when the inequality starts with a number, keep the order carefully.

Correct answer: A. (x=-3)

(-3) is included because of (\leq), while (1) is not included because of (<). Check open and closed endpoints separately.

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

From (10-5x\geq 0), (-5x\geq -10), so (x\leq 2). The boundary (2) is included because the sign was (\geq).

Correct answer: A. (x\geq 0)

A closed circle includes (0), and right-side shading shows numbers greater than (0). Therefore (x\geq 0) is correct.

Correct answer: A. (3x\geq 18)

Not less than means greater than or equal to (18). In word problems, match phrases correctly with inequality signs.