Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. (x\ge -4)

Dividing (-2x\le 8) by (-2) reverses the sign and gives (x\ge -4). This sign reversal is a very common exam mistake.

Correct answer: A. (x>3)

First subtract (1) to get (5x>15), then divide by (5) to get (x>3). Removing the constant term first makes solving easier.

Correct answer: A. (x\le 4)

Adding (7) gives (4x\le 16), and dividing by (4) gives (x\le 4). Dividing by positive (4) keeps the sign unchanged.

Correct answer: A. (x<3)

From (6-2x>0), we get (-2x>-6), and dividing by (-2) gives (x<3). Reverse the sign when dividing by a negative number.

Correct answer: A. (x\le 2)

The direct answer is option A: x≤2. Start with 7-3x≥1. Subtract 7 from both sides: -3x≥-6. Divide by -3; because the divisor is negative, reverse the sign: x≤2. A is correct. Check the boundary x=2: 7-3(2)=1, so 1≥1 is true and 2 must be included. B, x≥2, has the wrong direction caused by failing to reverse the sign. C, x≤-2, has the wrong boundary and does not describe the result; for example, x=0 satisfies the original inequality since 7≥1, but C excludes 0. D, x≥-2, also has the wrong direction and includes x=3, which gives 7-9=-2≥1, false. The minus coefficient of x is the key detail. Memory cue: when isolating a variable by dividing by a negative coefficient, turn ≥ into ≤.

Correct answer: A. (x\le 4)

Subtracting (x) and (5) from both sides gives (x\le 4). Keep variable terms on one side and constants on the other.

Correct answer: A. x > 4

The governing concept is solving a linear inequality containing the variable on both sides. Begin with 3x - 2 > x + 6. Subtract x from both sides, which preserves the inequality direction because only the same quantity is being subtracted, to get 2x - 2 > 6. Add 2 to both sides and obtain 2x > 8. Dividing by the positive number 2 gives x > 4. Thus option A is correct. A useful check is x = 5: the original inequality becomes 13 > 11, which is true. A value such as x = 3 gives 7 > 9, which is false. Option B reverses the solution region, while C and D result from incomplete or incorrect simplification.

Correct answer: A. x ≥ 4

This is an algebraic solution of a linear inequality in one variable. First subtract 2x from both sides: 2x + 3 ≥ 11. Next subtract 3: 2x ≥ 8. Dividing by the positive number 2 does not reverse the inequality sign, so x ≥ 4. Therefore option A is correct, and the solution set includes 4 and every number greater than 4. Option B incorrectly reverses the direction without a negative division. Options C and D confuse the constant 7 that may appear during an incomplete rearrangement with the actual boundary value. A quick check confirms the result: x = 4 gives 19 ≥ 19, while x = 5 gives 23 ≥ 21.

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

Subtracting \(3x\) from both sides gives \(2x-4<2\). Adding 4 to both sides gives \(2x<6\), and dividing by the positive number 2 gives \(x<3\). The inequality sign does not reverse because the divisor is positive. Exam tip: the sign reverses only when an inequality is multiplied or divided by a negative number, so \(x>3\) is incorrect.

Correct answer: A. (x\ge 4)

(9-x\le 5) gives (-x\le -4), and dividing by (-1) gives (x\ge 4). Reverse the sign when dividing by a negative number.

Correct answer: A. (x>6)

Multiplying both sides of (\frac{x}{2}>3) by (2) gives (x>6). Multiplication by positive (2) keeps the sign unchanged.

Correct answer: A. (x\le 12)

Multiplying both sides by (3) gives (x\le 12). When the denominator is positive, the sign does not change.

Correct answer: A. (x\ge 9)

Multiplying by (2) gives (x+1\ge 10), and subtracting (1) gives (x\ge 9). Solving step by step reduces mistakes.

Correct answer: A. (x<7)

Multiplying by (5) gives (x-2<5), and adding (2) gives (x<7). A positive denominator keeps the inequality sign unchanged.

Correct answer: A. (x>6)

Multiplying by (3) gives (2x>12), and dividing by (2) gives (x>6). Remove fractions to form a simple linear inequality.

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

Expanding the bracket gives \(2x+6>14\). Subtracting 6 from both sides results in \(2x>8\), and dividing by 2 gives \(x>4\). Therefore, option A is correct. Option B is incorrect because division by the positive number 2 does not reverse the inequality sign. Exam tip: Reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. (x\le 6)

From \(3(x-2)\le 12\), we get \(x-2\le 4\), so \(x\le 6\). Dividing by positive (3) does not change the sign.

Correct answer: A. (x\ge 3)

Opening brackets gives \(4x+4\ge 2x+10\), so \(2x\ge 6\) and \(x\ge 3\). Simplify all terms first.

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

Expanding the brackets gives \(5x-5<3x+7\). Subtracting \(3x\) from both sides yields \(2x-5<7\), or \(2x<12\). Dividing by the positive number \(2\) gives \(x<6\), so option A is correct. Option B is wrong because the inequality sign is reversed even though the divisor is positive. Exam tip: reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. x > 4

Expanding the brackets gives 6x − 2 > 4x + 6. Subtracting 4x from both sides gives 2x − 2 > 6, and adding 2 gives 2x > 8. Dividing by the positive number 2 yields x > 4, so option A is correct. Option B has the wrong direction; division by a positive number does not reverse an inequality. Exam tip: the inequality sign reverses only when multiplying or dividing by a negative number.

Correct answer: A. ((-\infty,5))

The direct answer is A: (-∞,5). The inequality x<5 includes every real number smaller than 5 and excludes 5 itself. In interval notation, the left side continues without end toward negative infinity, written −∞; infinity is never an included endpoint, so it uses a parenthesis. The right endpoint 5 is also excluded because the sign is strict <, so it also uses a parenthesis. Option A is correct. Option B includes 5 with a square bracket, which would mean x≤5. Option C starts at 5 and includes numbers greater than 5, the opposite direction. Option D also represents numbers greater than 5 and excludes 5. Memory cue: x<5 means shade left of 5 and leave 5 open.

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

In (x\ge -2), (-2) is included, so ([-2,\infty)) is correct. Infinity always takes a round bracket.

Correct answer: A. [4, ∞)

Adding 1 to both sides gives 2x ≥ 8. Dividing by 2 gives x ≥ 4, so 4 is included and the solution interval is [4, ∞).

Correct answer: A. (4)

(x-1>2) gives (x>3), so the smallest natural number is (4). A strict inequality does not include the boundary value.

Correct answer: A. (4)

(x+2\le 6) gives (x\le 4), and the positive integers are (1,2,3,4). Do not count (0) as a positive integer.