Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. (x\le 13)

The direct answer is A: \(x\le13\). Begin with \(x-4\le9\). To remove -4 from the left side, add 4 to both sides: \(x-4+4\le9+4\). This simplifies to \(x\le13\). Addition preserves the direction of an inequality, so the symbol does not change. Option A is correct and includes x=13 because the original sign is \(\le\). Option B, \(x\ge13\), reverses the relation and would include values larger than 13, which do not work; for x=14, the original statement becomes 10\le9, false. Option C, \(x<13\), wrongly excludes x=13, which actually gives 9\le9. Option D, \(x>13\), is also the wrong direction and excludes all valid smaller values. Important rule: adding or subtracting the same number does not reverse the sign; multiplying or dividing by a negative number does. Exam cue: keep the inequality sign unchanged when shifting terms by addition or subtraction.

Correct answer: A. x ≥ 3

The governing concept is that subtracting the same quantity from both sides preserves an inequality, and division by a positive number preserves its direction. Starting with 5x + 2 ≥ 17, subtract 2 from both sides to get 5x ≥ 15. Divide by 5, which is positive, and obtain x ≥ 3. The equality sign remains because x = 3 gives 5(3) + 2 = 17, so the original statement is true at the boundary. Therefore option A is correct. Option C incorrectly changes the inclusive relation to a strict one and excludes x = 3. Options B and D reverse the direction, placing the solution on the wrong side of the boundary. In interval notation, the solution is [3, ∞).

Correct answer: A. \(x<4\)

Adding 7 to both sides gives \(4x<16\). Dividing by 4 yields \(x<4\), and the inequality sign remains unchanged because 4 is positive. Therefore, option A is correct. The distractor \(x\le4\) is incorrect because the original inequality is strict and does not include equality. In an exam, always check whether division by a negative number requires reversing the inequality sign.

Correct answer: A. (x\ge 3)

Subtracting (7) gives (-2x\le -6), then division by (-2) reverses the sign to (x\ge 3). Be careful with a negative coefficient.

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

For the inequality \(2x+3<x+8\), subtracting \(x\) from both sides gives \(x+3<8\). Subtracting \(3\) from both sides then gives \(x<5\), so option A is correct. Option B incorrectly reverses the inequality sign; no multiplication or division by a negative number has occurred. Exam tip: Reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. \\(x\ge 3\\)

Subtracting \\(2x\\) from both sides gives \\(4x-1\ge 11\\). Adding 1 to both sides gives \\(4x\ge 12\\), and dividing by 4 gives \\(x\ge 3\\). Since 4 is positive, the inequality sign does not change. Remember that \\(x=3\\) is also included, so \\(x>3\\) is incorrect.

Correct answer: A. \\(x>3\\)

Dividing both sides by the positive number 3 gives \\(x+2>5\\). Subtracting 2 from both sides gives \\(x>3\\), so option A is correct. Option B incorrectly reverses the inequality sign; division by a positive number does not reverse it. Exam tip: Reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. x ≤ 9

The governing concept is equivalence in solving a linear inequality: the same operation may be applied to both sides, and division by a positive number leaves the inequality sign unchanged. Divide 2(x − 5) ≤ 8 by 2 to obtain x − 5 ≤ 4. Adding 5 to both sides gives x ≤ 9. Therefore option A is correct. The endpoint 9 must be included because the original relation is ≤; direct substitution gives 2(9 − 5) = 8, which is true. Options B and D place the solution on the wrong side of 9, while option C incorrectly excludes the valid boundary value. The complete solution is every real number less than or equal to 9.

Correct answer: A. (x\ge 9)

The direct answer is A: \(x\ge9\). Start with \((x-1)/4\ge2\). Multiply both sides by 4. Since 4 is positive, the inequality direction remains unchanged: \(x-1\ge8\). Add 1 to both sides: \(x\ge9\). Thus A is correct, and x=9 is included because the original sign is \(\ge\); substitution gives \((9-1)/4=2\). Option B, \(x\le9\), reverses the direction and is false for x=10, which satisfies the original inequality. Option C, \(x>9\), wrongly excludes the valid boundary x=9. Option D, \(x<9\), has both the wrong direction and excludes valid larger values. The denominator 4 is positive, so no sign reversal occurs. If the multiplier or divisor had been negative, the sign would have to reverse. Memory cue: positive multiplication keeps the sign; negative multiplication flips it.

Correct answer: A. \(x\le 7\)

Multiplying both sides by 5 gives \(2x+1\le 15\). Thus, \(2x\le 14\), so \(x\le 7\). Since 5 is positive, the inequality sign remains unchanged. Option B reverses the sign incorrectly, while option C wrongly excludes \(x=7\). In an exam, check whether the multiplier is positive or negative before deciding whether to reverse the inequality sign.

Correct answer: A. (x>-8)

Multiplying both sides by (-4) reverses the sign and gives (x>-8). Reverse the sign when multiplying by a negative number.

Correct answer: A. (x<3)

The direct answer is A: \(x<3\). Solve step by step: \(9-3x>0\). Subtract 9 from both sides to get \(-3x>-9\). Now divide by -3. Because the divisor is negative, the inequality sign must reverse: \(x<3\). Therefore A is correct. A direct check confirms it: if x=2, then \(9-3(2)=3>0\). Option B, \(x>3\), has the reversed direction and fails for x=4, giving \(9-12=-3>0\), which is false. Option C, \(x\le3\), incorrectly includes x=3; at x=3 the expression is 0, and 0>0 is false. Option D, \(x\ge3\), also includes the invalid boundary and values above it. The strict sign > means equality is not allowed. The key rule is that multiplying or dividing by a negative number reverses the inequality. Memory cue: “negative crossing flips the sign.”

Correct answer: A. x ≤ 7

The governing concept is collecting variable terms on one side and constants on the other without changing the inequality through addition or subtraction. From 2x + 5 ≤ x + 12, subtract x from both sides to get x + 5 ≤ 12. Next subtract 5 from both sides, giving x ≤ 7. Therefore option A is correct. Equality is included because x = 7 changes the original statement to 14 + 5 ≤ 7 + 12, or 19 ≤ 19, which is true. Options C and D wrongly exclude or reverse the boundary, whereas B gives the opposite direction. All real numbers up to and including 7 satisfy the inequality.

Correct answer: A. x > 6

The governing concept is that multiplying or dividing an inequality by a negative number reverses its direction. Begin with 10 − x < 4. Subtract 10 from both sides to obtain −x < −6. To isolate x, divide by −1; because the divisor is negative, the sign reverses, giving x > 6. Thus option A is correct. The reversal is essential: forgetting it would produce x < 6, option B. Options C and D also mishandle the strict sign or direction. A quick check confirms the result: x = 7 gives 3 < 4, true, while x = 5 gives 5 < 4, false. Hence precisely numbers greater than 6 satisfy the inequality.

Correct answer: A. x ≥ 3

The governing concept is algebraic equivalence for inequalities: expand brackets, collect like terms, and divide by a positive coefficient without reversing the sign. Expanding gives 4x + 4 ≥ 2x + 10. Subtract 2x from both sides to obtain 2x + 4 ≥ 10, and subtract 4 to get 2x ≥ 6. Dividing by positive 2 yields x ≥ 3, so option A is correct. The equality is retained because x = 3 makes both sides equal to 16. Option C incorrectly excludes this valid boundary, while B and D reverse the direction. The complete solution is every real number at least 3.

Correct answer: A. x < 7

The governing concept is that equivalent algebraic operations preserve a linear inequality, except that division by a negative number would reverse its sign. First expand the bracket: 5x − 10 < 3x + 4. Subtract 3x from both sides to get 2x − 10 < 4. Add 10, giving 2x < 14. Since the divisor 2 is positive, division produces x < 7 without reversing the sign. Hence option A is correct. The strict inequality excludes 7: when x = 7, the original statement becomes 5(5) < 25, or 25 < 25, which is false. Option B reverses the direction, and options C and D incorrectly include the endpoint.

Correct answer: A. (3)

The inequality gives (x<4), so the greatest integer is (3). A strict sign does not include the boundary value.

Correct answer: A. (5)

Dividing by (2) gives (x\ge 5), so the least integer is (5). The equality sign includes the boundary value.

Correct answer: A. (3, ∞)

The governing concept is the algebraic solution of a strict linear inequality and its representation in interval notation. Start with 3x + 1 > 10. Subtract 1 from both sides to obtain 3x > 9, and divide by the positive number 3, giving x > 3. Because the inequality is strict, x = 3 is not included in the solution set, so a round or open bracket is required at 3. The solution continues without bound toward positive infinity, which is always written with a parenthesis. Therefore option A, (3, ∞), is correct. Option B incorrectly includes 3, while options C and D describe values less than 3.

Correct answer: A. (−∞, 2]

The governing concept is converting an algebraic inequality into interval notation while correctly treating its boundary. Starting with 4x − 8 ≤ 0, add 8 to both sides to obtain 4x ≤ 8. Divide by the positive number 4, giving x ≤ 2. Numbers less than 2 and the number 2 itself are included, while numbers greater than 2 are excluded. Therefore the interval is (−∞, 2], which is option A. The square bracket at 2 records inclusion because the relation is ≤. Option B wrongly excludes 2, and C and D describe values to the right of 2 rather than to its left. Infinity always uses a parenthesis, never a square bracket.

Correct answer: A. x = 4

The governing concept is solving a linear inequality while preserving the correct order relation. Starting with 5 − x ≥ 2, subtract 5 from both sides to obtain −x ≥ −3. Multiplying or dividing an inequality by −1 reverses its sign, so x ≤ 3. Therefore every value less than or equal to 3 satisfies the inequality, while values greater than 3 do not. Testing the options confirms this: x = 3 gives 2 ≥ 2, x = 2 gives 3 ≥ 2, and x = 1 gives 4 ≥ 2. For x = 4, the left side is 1, and 1 ≥ 2 is false. Hence option A is the only value that does not satisfy the inequality. The other options are distractors because they lie within the solution set.

Correct answer: A. Subtract (9) from both sides

To isolate (x), subtract (9) from both sides. Choosing the inverse operation saves time in exams.

Correct answer: A. Dividing by −5 reverses the sign

The governing concept is the order property of inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign. In −5x ≤ 20, division by −5 gives x ≥ −4, not x ≤ −4. The reversal is necessary because the order of numbers changes when both are multiplied by a negative quantity; for example, 2 > 1 becomes −2 < −1. Adding or subtracting the same number from both sides does not reverse the sign, so options C and D state incorrect rules. Option B is also incorrect because it ignores the negative divisor. Thus option A correctly identifies the rule. A quick check confirms the result: x = −4 gives −5(−4) = 20, and larger x values make −5x smaller, so they satisfy −5x ≤ 20.

Correct answer: A. (x\ge 4)

Dividing both sides by (3) gives (4\le x), that is (x\ge 4). Write the final answer starting with (x) in standard form.

Correct answer: B. \(x<3\)

Divide both sides by \(6\) (positive number, so the inequality sign does not change): \(18>6x\Rightarrow 18/6>x\Rightarrow 3>x\). Hence \(x<3\).