Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. (3)

(3x\ge 7) gives (x\ge \frac{7}{3}), so the least integer is (3). Take the first integer above the boundary.

Correct answer: A. 3

Subtracting 5 from both sides of \(5-x>1\) gives \(-x>-4\). Multiplying both sides by \(-1\) reverses the inequality sign, so \(x<4\). The greatest integer less than 4 is 3; therefore, 3 is correct. Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative number.

Correct answer: C. \(x>2\)

Substitute \(x=3\):
- \(x<2\) becomes \(3<2\) (false).
- \(x=3\) is an equality, not the given inequality type.
- \(x>2\) becomes \(3>2\) (true).
- \(x\le 0\) becomes \(3\le 0\) (false).
So \(x>2\) is the true inequality.

Correct answer: A. (x\ge 0)

The direct answer is option A: x≥0. Begin with x+7≥7. Subtract 7 from both sides: x≥0. Subtraction does not change the inequality direction. A is correct and includes 0, because 0+7=7 and 7≥7 is true. B, x≤0, reverses the direction without a reason and wrongly excludes positive solutions such as x=1. C, x≥14, comes from an incorrect operation; the 7 is removed by subtracting 7, not by adding 7 to obtain 14. It also wrongly excludes x=1. D, x≤14, has the wrong direction and includes negative values such as -1, for which -1+7=6≥7 is false. Equality matters: the original sign is ≥, so the boundary value 0 is included. On a number line, shade 0 and everything to its right. Memory cue: subtract the constant, preserve the sign, and include the endpoint for ≥ or ≤.

Correct answer: A. (x<3)

Adding (4) to both sides gives (x<3). Even with negative numbers, addition does not change the sign.

Correct answer: A. \(x>-8\)

Subtracting 6 from both sides of \(x+6>-2\) gives \(x>-2-6=-8\). Therefore, the correct solution is \(x>-8\). The inequality sign does not change when the same number is subtracted from both sides; it reverses only when multiplying or dividing by a negative number. Exam tip: move the constant term to the other side and change its sign.

Correct answer: A. Dividing by (-3) reverses the sign

In (-3x>12), we divide by the negative number (-3), so the sign reverses. This rule is very important in inequalities.

Correct answer: A. x ≤ 6

The governing concept is solving a one-variable linear inequality while preserving its direction. Begin with 8 + 2x ≤ 20. Subtract 8 from both sides to isolate the variable term: 2x ≤ 12. Divide both sides by the positive number 2, obtaining x ≤ 6. Since the divisor is positive, the inequality sign remains unchanged; therefore option A is correct. Option B reverses the sign incorrectly. Options C and D use 14, which comes from subtracting in the wrong direction or failing to simplify the variable coefficient. Checking the boundary value x = 6 gives 8 + 12 = 20, and a smaller value such as x = 0 gives 8 ≤ 20, confirming the solution region.

Correct answer: A. (x+5>12)

Let the number be (x). Adding (5) gives (x+5), and greater than means (>). Convert words into algebra first.

Correct answer: A. (x<9)

The statement gives (2x<18), and dividing by (2) gives (x<9). Dividing by a positive number does not change the sign.

Correct answer: A. (x-30\ge 70)

After spending, the amount is (x-30), and at least (70) means (\ge 70). In words, at least is written using (\ge).

Correct answer: A. \(x\le 25\)

The cost of four pens is \(4x\) rupees. “Not more than 100” means \(4x\le 100\). Dividing both sides by the positive number 4 gives \(x\le 25\), so option A is correct. In exams, remember that “not more than” indicates the \(\le\) sign, not the strict sign \(<\).

Correct answer: A. (x\ge 8)

The statement gives (x-8\ge 0), and adding (8) gives (x\ge 8). Use (\ge) for greater than or equal to.

Correct answer: A. (x\le 3)

Subtracting (2) from (7x+2\le 23) gives (7x\le 21), and dividing by (7) gives (x\le 3). Division by a positive number does not change the sign.

Correct answer: A. (x>-3)

Subtracting (5) gives (-4x<12), and dividing by (-4) gives (x>-3). The inequality sign reverses when dividing by a negative number.

Correct answer: A. (x<5)

Multiplying by (4) gives (x+3<8), and subtracting (3) gives (x<5). The sign stays the same when clearing a positive denominator.

Correct answer: A. (x\ge 3)

The direct answer is option A, x\ge3. Begin with 2x+1\le5x-8. Subtract 2x from both sides: 1\le3x-8. Add 8 to both sides: 9\le3x. Divide by positive 3: 3\le x, which is the same as x\ge3. Option A is correct. Option B, x\le3, reverses the direction without a valid reason; no negative number was used. Option C, x\ge-3, has the wrong boundary because the correct division gives 3, not -3. Option D, x\le-3, has both the wrong sign direction and wrong boundary. Check x=3: left side is 7 and right side is 7, so equality is allowed. Check x=4: 9\le12 is true, while x=2 gives 5\le2, false. Memory cue: only multiplication or division by a negative number reverses an inequality.

Correct answer: A. (x\le 13)

The statement gives (x+3\le 16), and subtracting (3) gives (x\le 13). Not more than means (\le).

Correct answer: A. (x\ge 4)

Subtracting (9) from (9-2x\le 1) gives (-2x\le -8), and dividing by (-2) gives (x\ge 4). The sign reverses when dividing by a negative number.

Correct answer: A. x ≤ 8

The governing concept is preservation of an inequality while performing equivalent algebraic operations. Start with 4x − 9 ≤ 2x + 7. Subtract 2x from both sides to obtain 2x − 9 ≤ 7. Add 9 to both sides, giving 2x ≤ 16. Dividing by the positive number 2 does not reverse the inequality sign, so x ≤ 8. Therefore option A is correct. The endpoint 8 is included because the original symbol is ≤; substituting x = 8 gives 32 − 9 = 23 and 16 + 7 = 23. Options B and D reverse the direction incorrectly, while option C excludes the valid endpoint and uses a strict inequality.

Correct answer: A. x ≥ 4

The governing idea is to perform the same operation on both sides while preserving the order. Starting from 15 ≤ 4x - 1, add 1 to both sides, obtaining 16 ≤ 4x. Dividing by the positive number 4 does not reverse the inequality, so 4 ≤ x. Written with x on the left, this is x ≥ 4; hence option A is correct. The equality sign remains because the original inequality includes equality. Option B reverses the direction, while options C and D either omit the boundary or point in the wrong direction. Checking x = 4 gives 15 ≤ 15, confirming that 4 is included.

Correct answer: A. x = 5

The governing concept is solving an inequality and then testing the proposed values against the resulting condition. Starting with 4x - 6 > 10, add 6 to both sides to obtain 4x > 16. Dividing by the positive number 4 does not change the inequality direction, so x > 4. Among the choices, x = 5 is the only value greater than 4. The value x = 4 is not accepted because the inequality is strict; equality would not satisfy >. The values 3 and 2 are below the required boundary. Direct substitution confirms the result: 4(5) - 6 = 14, and 14 > 10. Therefore option A is the unique correct answer, while the other options fail the original inequality.

Correct answer: A. x = 4

The governing concept is solving a linear inequality and observing that division by a negative number reverses its sign. From 12 - 3x ≤ 0, subtract 12 from both sides to get -3x ≤ -12. Dividing by -3 changes ≤ into ≥, giving x ≥ 4. Of the listed values, only x = 4 satisfies this condition; 3, 2 and 1 are all less than 4. Equality is allowed because the original sign is ≤, so the endpoint x = 4 belongs to the solution set. Substitution provides a direct check: 12 - 3(4) = 12 - 12 = 0, and 0 ≤ 0 is true. Therefore option A is correct, whereas each other option makes the original inequality false.

Correct answer: C. x ≥ 18

The governing concept is solving a linear inequality while preserving its direction. Start with x/6 − 1 ≥ 2. Add 1 to both sides, giving x/6 ≥ 3. Multiply both sides by 6. Because 6 is positive, the inequality sign remains unchanged: x ≥ 18. Thus every real number greater than or equal to 18 satisfies the original inequality, and 18 itself is included because the symbol is ≥. Option C is correct. Option A and option D reverse the direction incorrectly, while option B excludes 18 and uses a strict inequality. A sign reversal would be required only if we multiplied or divided by a negative number, which does not happen here. On a number line, the answer is shown by a closed point at 18 with shading to the right.

Correct answer: D. {x : x ≥ −5}

The governing concept is solving a linear inequality and expressing its answer in set-builder notation. Begin with 7x + 2 ≥ 3x − 18. Subtract 3x from both sides to obtain 4x + 2 ≥ −18. Then subtract 2, giving 4x ≥ −20. Dividing by the positive number 4 preserves the inequality sign, so x ≥ −5. In set notation, the solution is {x : x ≥ −5}, which includes −5 because equality is permitted by the original ≥ sign. Therefore option D is correct. Option A reverses the direction, option B incorrectly excludes the endpoint, and option C uses the wrong strict direction. A useful check is x = −5: both sides become −33, so equality holds and −5 must be included.