Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. \\(x\le 15\\)

Adding 11 to both sides of \\(x-11\le 4\\) gives \\(x\le 15\\), so option A is correct. Option B reverses the inequality without justification, while options C and D incorrectly remove the possibility of equality. Exam tip: the inequality sign changes direction only when multiplying or dividing by a negative number, not when adding a number.

Correct answer: A. (x<7)

Dividing both sides by (6) gives (x<7). Dividing by a positive number keeps the sign unchanged.

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

The direct answer is option A, x\le -7. Start with -3x\ge 21. Divide both sides by -3. Whenever an inequality is divided or multiplied by a negative number, its sign reverses, so \ge becomes \le. Thus x\le 21/(-3)=-7. Option A is correct and includes -7 because the original sign is inclusive. Option B, x\ge -7, forgets to reverse the sign. Option C, x<-7, reverses the sign but wrongly removes equality; x=-7 actually satisfies the original inequality because -3(-7)=21. Option D, x>-7, has both the wrong direction and the wrong boundary. A quick check confirms the result: x=-8 gives 24\ge21, while x=0 gives 0\not\ge21. Memory cue: dividing an inequality by a negative flips the sign.

Correct answer: A. x > 6

Subtracting 6 from both sides gives 4x > 24. Dividing both sides by the positive number 4 gives x > 6, so option A is correct. Option C is incorrect because equality at x = 6 is not included in the original strict inequality. Exam tip: reverse the inequality sign only when multiplying or dividing by a negative number; division by positive 4 does not reverse it.

Correct answer: A. (x\le 4)

Adding (5) to both sides gives (9x\le 36), so (x\le 4). Division by a positive coefficient does not change the sign.

Correct answer: A. x < 4

The governing rule is that dividing or multiplying an inequality by a negative number reverses its sign. Start with 18 − 4x > 2. Subtract 18 from both sides: −4x > −16. Now divide both sides by −4. Because −4 is negative, the greater-than sign changes to less-than, giving x < 4. Thus option A is correct. The boundary value x = 4 is not included because the original sign is strict: substituting 4 gives 18 − 16 = 2, not a value greater than 2. Option B is the common error of forgetting sign reversal. Options C and D incorrectly use a non-strict sign, and D also has the wrong direction.

Correct answer: A. x ≥ 3

The governing concept is solving an inequality with a negative coefficient and reversing the sign when dividing by that coefficient. Begin with 21 − 7x ≤ 0. Subtract 21 from both sides to obtain −7x ≤ −21. Divide by −7; because the divisor is negative, the less-than-or-equal sign reverses to greater-than-or-equal. Therefore x ≥ 3, so option A is correct. The endpoint is included because the original relation contains equality. Checking x = 3 gives 21 − 21 = 0, which satisfies the inequality. Values larger than 3 make 21 − 7x negative and also satisfy it. Option B results from failing to reverse the sign, while C and D incorrectly exclude or reverse the boundary.

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

Subtracting \(x\) from both sides gives \(2x+10<22\). Subtracting 10 and then dividing by 2 gives \(2x<12\), so \(x<6\). Therefore, option A is correct. Option B incorrectly reverses the inequality sign; no multiplication or division by a negative number occurs here. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

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

Subtracting \(5x\) from both sides gives \(3x-3\ge 18\). Adding 3 to both sides yields \(3x\ge 21\), and dividing by 3 gives \(x\ge 7\). Therefore, option A is correct. Option B reverses the inequality without justification; the inequality sign changes direction only when multiplying or dividing by a negative number. Exam tip: Always check whether the number used for multiplication or division is negative.

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

Dividing both sides by the positive number 5 gives \(x+1>5\). Subtracting 1 from both sides yields \(x>4\), so option A is correct. Since the inequality is strict, \(x=4\) is not included; therefore, \(x\ge4\) is incorrect. Exam tip: The inequality sign does not change when multiplying or dividing by a positive number.

Correct answer: A. (x\le 10)

The direct answer is A: x≤10. Start with 3(x−6)≤12. Divide both sides by the positive number 3: x−6≤4. Add 6 to both sides: x≤10. The inequality sign does not reverse because division was by a positive number; it reverses only when multiplying or dividing by a negative number. Option A is correct. Option B reverses the direction without reason. Option C changes ≤ into < and wrongly excludes 10; substituting x=10 gives equality, so 10 must be included. Option D both reverses the direction and excludes the boundary. A quick check with x=10 gives 3(4)=12, which satisfies ≤. Memory cue: positive division keeps the sign; negative division flips it.

Correct answer: A. (x<\frac{27}{5})

Opening brackets gives (7x-14<2x+13), so (5x<27) and (x<\frac{27}{5}). A fractional answer can also be correct.

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

Expand the bracket: \(4(x+3)=4x+12\). So \(4x+12\ge 2x+20\). Subtracting \(2x\) gives \(2x+12\ge 20\); subtract 12 to get \(2x\ge 8\). Dividing by \(2\) (positive) preserves the inequality: \(x\ge 4\). The closest distractor \(x>4\) omits the equality case \(x=4\). Exam tip: always test the boundary value (here \(x=4\)) to confirm whether equality is allowed.

Correct answer: A. x < 30

The governing concept is isolating the variable in a linear inequality and noting that multiplication by a positive number preserves the inequality direction. Start with x/5 + 3 < 9. Subtract 3 from both sides to get x/5 < 6. Multiply both sides by 5. Since 5 is positive, the sign remains unchanged, and x < 30 follows. Therefore option A is correct. The value x = 30 is excluded because it makes the left side 30/5 + 3 = 9, giving equality rather than a strict less-than result. Options C and D incorrectly include the boundary, while B reverses the solution direction without a valid negative operation. Any number smaller than 30 can be checked and satisfies the inequality.

Correct answer: A. \(x\ge 11\)

Multiplying both sides by the positive number 2 does not reverse the inequality: \(x+5\ge 16\). Subtracting 5 from both sides gives \(x\ge 11\), so option A is correct. Option C is incorrect because \(x=11\) is also included in the solution set. Exam tip: the inequality sign reverses when both sides are multiplied or divided by a negative number, but not by a positive number.

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

Multiplying both sides by 3 does not reverse the inequality because 3 is positive: \(4x-1\le 15\). Adding 1 to both sides gives \(4x\le 16\), and dividing by 4 gives \(x\le 4\). Therefore, option A is correct. Option C is incorrect because \(x=4\) also satisfies the original inequality. Exam tip: Reverse the inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. (x\le -18)

Multiplying both sides by (-6) reverses the sign and gives (x\le -18). Change the direction in negative multiplication.

Correct answer: A. x > 17/2

The governing concept is clearing a positive denominator before isolating the variable. The inequality is (2x + 7)/4 > 6. Multiply both sides by 4. Because 4 is positive, the inequality direction remains unchanged: 2x + 7 > 24. Subtract 7 from both sides to obtain 2x > 17. Divide by the positive number 2, giving x > 17/2, or equivalently x > 8.5. Hence option A is correct. The endpoint x = 17/2 is excluded because the original relation is strict. Option B reverses the direction without justification, and C and D replace the strict sign with an inclusive one. A quick check with x = 9 gives 25/4 > 6, confirming the solution direction.

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

First isolate the term containing x. From \\(5x-10<15\\), add 10 to both sides, giving \\(5x<25\\). Next divide both sides by the positive number 5, so the inequality direction stays unchanged. The result is \\(x<5\\).

The solution contains every real number smaller than 5, but it does not contain 5 itself. In interval notation, an unbounded set extending left from 5 is written as \\((-\\infty,5)\\). The parenthesis at 5 is open because the inequality is strict. Therefore option A is correct. Option B would include 5 and would represent \\(x\\le5\\), while options C and D describe values greater than 5.

Correct answer: A. x ≥ 3

The governing concept is maintaining an equivalent inequality while isolating the variable. Begin with 12 ≤ x + 9. Subtract 9 from both sides: 12 − 9 ≤ x, so 3 ≤ x. This is conventionally written as x ≥ 3, which is option A. The equality is included because the original symbol is ≤. Indeed, x = 3 gives 12 ≤ 3 + 9, or 12 ≤ 12, which is true. Any value greater than 3 also satisfies the relation. Option B reverses the direction and describes values at most 3, while C and D incorrectly remove the equality and/or reverse the direction. Rewriting 3 ≤ x as x ≥ 3 is a change in order only, not a change in meaning.

Correct answer: B. x < 6

From \(30>5x\). Divide both sides by \(5\) (positive, so the inequality sign stays the same): \(6>x\), which is the same as \(x<6\).

Correct answer: A. x = 4

The governing concept is solving the inequality first and then testing the listed values. Starting with 10 − 2x ≥ 4, subtract 10 from both sides to obtain −2x ≥ −6. Dividing by −2 reverses the sign, so x ≤ 3. Therefore every value at most 3 satisfies the inequality, whereas x = 4 does not; option A is correct. Direct checking confirms this: for x = 4, 10 − 2(4) = 2, and 2 ≥ 4 is false. For x = 3, the result is 4, which satisfies equality. For x = 2 and x = 1, the results are 6 and 8, both at least 4. Thus the question’s wording asks for the one value outside the solution set, namely 4.

Correct answer: A. (4)

The inequality gives (x<5), so the greatest integer is (4). A strict sign does not include the boundary.

Correct answer: A. (8)

The direct answer is option A, x=8. Since 3x\ge24 and 3 is positive, divide both sides by 3 without changing the inequality sign: x\ge8. The question asks for the least possible integer. The first integer satisfying x\ge8 is 8, so option A is correct. Option B, 7, is too small because 3(7)=21, which is not at least 24. Option C, 9, does satisfy the inequality, since 27\ge24, but it is not the least possible value because 8 is smaller and also works. Option D, 24, also satisfies the inequality, but it is not the least value; it appears to confuse the right side with the answer. The symbol \ge includes the endpoint, so x=8 is allowed. Memory cue: solve the inequality first, then choose the smallest allowed integer.

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

The direct answer is option A: {0,1,2,3}. A whole number is a non-negative integer: 0,1,2,3,4, and so on. The condition x<4 means x must be smaller than 4, so list the whole numbers beginning at 0 and stop before 4: 0,1,2,3. Thus 4 is excluded because the inequality is strict. A is correct. B, {1,2,3,4}, wrongly omits 0 and includes 4. C, {0,1,2,3,4}, also includes 4, which is not allowed. D, {4,5,6}, contains numbers that are equal to or greater than 4, so none of them satisfies x<4. The word whole is important: whole numbers include 0, unlike positive counting numbers in conventions that begin with 1. The answer is finite here because the domain is restricted to whole numbers. Memory cue: for x<4, write the whole numbers from 0 up to, but not including, 4.