Algebraic Solution of Linear Inequalities in One Variable

Correct answer: B. (−∞, 4]

Subtracting 5x and adding 4 gives 4x ≤ 16. Dividing by 4 gives x ≤ 4, so the interval is (−∞, 4], including 4.

Correct answer: B. (−3, ∞)

From −4x − 1 < 11, we get −4x < 12. Dividing both sides by −4 reverses the inequality sign, giving x > −3. Since the inequality is strict, −3 is not included, so the interval is (−3, ∞). Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative number.

Correct answer: C. \(x=5\)

Starting with \(5x+2\ge 27\), subtracting 2 from both sides gives \(5x\ge 25\). Dividing by 5 gives \(x\ge 5\); the inequality sign remains unchanged because 5 is positive. Therefore, \(x=5\) is correct as it is the boundary value. The closest distractor, \(x=4\), gives \(22\ge 27\), which is false. Exam tip: the inequality sign reverses only when dividing or multiplying by a negative number.

Correct answer: B. (x=7)

From (-2x<-14), we get (x>7). Since the boundary (7) is not included, none of the given values should be correct.

Correct answer: C. \(x=7\)

Rearranging gives \(-2x\leq -14\). Dividing by \(-2\) reverses the inequality sign, so \(x\geq 7\). Of the listed values, only \(x=7\) satisfies this, and substituting it gives \(20-14=6\). Therefore, option C is correct.

Correct answer: B. All real (x)

The direct answer is B: all real values of x. An inequality compares two expressions. Here, subtracting 8x from both sides is allowed because the same quantity is removed from both sides, so the inequality sign does not change: 8x+3>8x−10 becomes 3>−10. This statement is always true because 3 is greater than −10, whatever real number x may be. Therefore every real number satisfies the inequality. Option A, no solution, is wrong because the reduced statement is true, not false. Option B is correct because x disappears and no restriction remains. Option C, x>3, is wrong because values such as x=0 also work. Option D, x<−10, is wrong because values such as x=5 also work. The key exam idea is that identical variable terms on both sides cancel. Memory cue: if the variable disappears and the remaining statement is true, all real numbers are solutions; if it is false, there is no solution.

Correct answer: C. No solution

Subtracting \\(5x\\) from both sides gives \\(-6 \ge 2\\), which is false. Therefore, no real number satisfies the inequality, so the solution set is empty. Options B and D incorrectly give bounds for \\(x\\), even though all \\(x\\)-terms cancel. Exam tip: after simplifying an inequality, a false statement means there is no solution.

Correct answer: A. When multiplying or dividing by a negative number

Multiplying or dividing by a negative number reverses order, so the sign flips. For example, \(-2x<6\Rightarrow x>-3\). Adding or subtracting the same number does not change it. Exam tip: check for a negative multiplier before writing the final inequality.

Correct answer: A. \(x<\frac{6}{a}\)

From \(ax+4>10\), we first obtain \(ax>6\). Since \(a<0\), dividing both sides by \(a\) reverses the inequality sign, giving \(x<\frac{6}{a}\). Option B ignores this sign reversal, while options C and D incorrectly include equality. Exam tip: always reverse the inequality sign when multiplying or dividing an inequality by a negative number.

Correct answer: B. 6x>42

Let the number be x. Six times the number is written as 6x, and ‘greater than 42’ requires the symbol >. Therefore, the correct inequality is 6x>42. Options A and D represent adding or subtracting 6, not multiplying by 6, while option C means less than 42. In exams, translate ‘times’ as multiplication and ‘greater than’ as >.

Correct answer: B. हल-समुच्चय संख्या रेखा पर एक अंतराल या किरण के रूप में दर्शाया जा सकता है।

A linear inequality in one variable usually gives an interval or a ray on the number line. For example, \(x>3\) contains every real number greater than 3. Two disconnected intervals are more typical of quadratic inequalities. Exam tip: an open circle means the boundary value is excluded.

Correct answer: B. \(x\leq 8\)

‘At most 35’ means the result is less than or equal to 35, so the inequality is \(4x+3\leq 35\). Subtracting 3 from both sides gives \(4x\leq 32\), and dividing by the positive number 4 gives \(x\leq 8\). Therefore, option B is correct. Exam tip: use \(\leq\) for ‘at most’ and \(\geq\) for ‘at least’.

Correct answer: C. 6

Let the number of pens be x. The remaining money is \(120-15x\), and the saving condition gives \(120-15x\geq 30\). Thus, \(-15x\geq -90\), so \(x\leq 6\). Since the number of pens must be a whole number, the maximum is 6. Buying 7 pens would leave only 15 rupees, which is less than the required 30 rupees. Exam tip: When an inequality is divided by a negative number, its sign is reversed.

Correct answer: D. 33

Let \(x\) be the number of additional passengers. The capacity condition is \(x+19\leq52\), so \(x\leq33\). Therefore, at most 33 more passengers can sit. Option 34 is incorrect because it would make the total 53 passengers, exceeding the capacity of the bus. In an exam, remember that “at most” translates to \(\leq\).

Correct answer: B. \(x\geq 6\)

Expanding the brackets gives \(3(2x-5)-2(x+1)=6x-15-2x-2=4x-17\). Thus, \(4x-17\geq 7\), so \(4x\geq 24\) and \(x\geq 6\). Option A reverses the inequality, while options C and D incorrectly include or exclude the boundary value 6. Exam tip: When dividing an inequality by a positive number, its sign does not reverse.

Correct answer: A. \\(x<5\\)

Expanding the brackets gives \\(10-2x+3>3x-12\\), or \\(13-2x>3x-12\\). Adding \\(12\\) to both sides gives \\(25>5x\\); dividing by the positive number 5 yields \\(x<5\\). The inequality sign does not reverse because the divisor is positive. Option B gives the opposite direction, while C and D incorrectly include equality. Exam tip: reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: B. \(x\leq 12\)

Using the common denominator 12, we get \(\frac{8x-3x}{12}\leq 5\), so \(\frac{5x}{12}\leq 5\). Multiplying both sides by the positive number \(\frac{12}{5}\) does not reverse the inequality, giving \(x\leq 12\). Hence, option B is correct. Option C is incorrect because it excludes \(x=12\), even though equality is included in the original inequality. Exam tip: multiplying or dividing an inequality by a positive number leaves its direction unchanged.

Correct answer: A. \(x>16\)

The least common denominator is 6. Multiplying both sides by the positive number 6 does not change the inequality sign: \(5x+6>2x+54\). Thus, \(3x>48\), so \(x>16\). Therefore, option A is correct. The distractor \(x\geq16\) is incorrect because substituting \(x=16\) makes both sides equal, whereas the inequality is strict. Exam tip: when an inequality is multiplied or divided by a negative number, its sign must be reversed.

Correct answer: A. (-11\leq x<9)

Multiplying by (2) gives (-10\leq x-1<8). Adding (1) gives (-9\leq x<9), so check endpoints carefully.

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

In the inequality \(-3\le x<4\), \(-3\) is included because the symbol is \(\le\), while \(4\) is excluded because the symbol is \(<\). Thus, all integers from \(-3\) through \(3\) are solutions, so option C is correct. Option B is incorrect because it includes \(4\). In an exam, remember that a closed endpoint is included and an open endpoint is excluded.

Correct answer: D. 4

From \(3x+4<19\), we get \(3x<15\), so \(x<5\). Since positive integers start at 1, the possible values are \(x=1,2,3,4\), giving 4 solutions. The distractor 5 is incorrect because \(x=5\) gives \(3(5)+4=19\), which does not satisfy the strict inequality \(<19\). Exam tip: the boundary value is excluded in a strict inequality.

Correct answer: B. 7

Simplifying \\(11x+6\geq 7x+34\\) gives \\(4x\geq 28\\), so \\(x\geq 7\\). Therefore, the least integer solution is 7. The value 6 is not allowed because it does not satisfy the inequality, while 8 and 9 are valid solutions but are not the least. In an exam, check the smallest integer at the boundary after solving the inequality.

Correct answer: C. Because \(x\leq 7\)

Solving the inequality gives \(2x-5\leq 9\), so \(2x\leq 14\) and therefore \(x\leq 7\). Thus, \(x=7\) belongs to the solution set because \(\leq\) means ‘less than or equal to’, which includes equality. Option A is incorrect because \(x<7\) excludes \(x=7\). Exam tip: The boundary value is included when the inequality sign is \(\leq\) or \(\geq\).

Correct answer: C. 4

Solving the inequality gives 4x + 1 > 17 ⇒ 4x > 16 ⇒ x > 4. Hence, the boundary point of the solution set is 4. Since the inequality is strict (>), 4 is not included in the solution set and is represented by an open point. Exam tip: After isolating x, the resulting value is the boundary point; check whether the inequality sign is strict or inclusive to decide whether the point is excluded or included.

Correct answer: D. \((-\infty,8]\)

Subtracting 7 from both sides gives \(-2x\geq -16\). Dividing by \(-2\) reverses the inequality sign, so \(x\leq 8\). Therefore, all numbers less than or equal to 8 are solutions, giving the interval \((-\infty,8]\). Option B excludes 8, so it is incorrect. Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative number.