Algebraic Solution of Linear Inequalities in One Variable

Correct answer: C. (x\ge 14)

The perimeter of a rectangle is \\(2(\text{length}+\text{breadth})\\). Here the length is 8 cm and the breadth is \\(x-2\\) cm. The phrase “at least 40 cm” means the perimeter may be 40 or greater, so the inequality sign is \\(\ge\\), not a strict greater-than sign.

Substitution gives \\(2(8+x-2)\ge40\\), so \\(2(x+6)\ge40\\). Dividing by 2 gives \\(x+6\ge20\\), and subtracting 6 gives \\(x\ge14\\). Therefore, option C is correct. The breadth condition also requires \\(x-2>0\\), which is automatically satisfied by \\(x\ge14\\). Option D would be too weak because values from 10 to 14 do not meet the perimeter requirement.

Correct answer: D. (x<12)

(500-25x>200) gives (-25x>-300), so (x<12). Reverse the sign when dividing by a negative coefficient.

Correct answer: A. ([-4,8))

(-3\le 2x+5<21) gives (-8\le 2x<16), so ([-4,8)). Write open and closed endpoints correctly in interval form.

Correct answer: B. (-3\le x<5)

We get (-10<-2x\le 6), and division by (-2) reverses direction, so (-3\le x<5). Reverse both signs when dividing by a negative.

Correct answer: C. −1 ≤ x < 3

Treat the compound inequality as two conditions that must hold simultaneously. Since 6 is positive, multiply all three parts by 6 without reversing either sign: 0 < 9 − 3x ≤ 12. From 0 < 9 − 3x, subtract 9 to get −9 < −3x, and divide by −3, reversing the sign: x < 3. From 9 − 3x ≤ 12, subtract 9 to get −3x ≤ 3, and divide by −3: x ≥ −1. Combining the conditions gives −1 ≤ x < 3, so option C is correct. The lower endpoint is included because x = −1 makes the fraction 2, while x = 3 is excluded because it makes the fraction 0.

Correct answer: D. (-\frac{7}{5}\le x<\frac{13}{5})

(-8\le 5x-1<12) gives (-7\le 5x<13), so (-\frac{7}{5}\le x<\frac{13}{5}). Preserve boundary signs at every step.

Correct answer: A. (x\ge \frac{19}{7})

Multiplying by (12) gives (4x+8+3x-15\ge 12), so (x\ge \frac{19}{7}). Use the LCM to clear denominators.

Correct answer: B. (x\ge -1)

Multiplying by (10) gives (4x-6-5x-5\le -10), so (x\ge -1). Change signs of both terms after a subtracted fraction.

Correct answer: C. (x<-\frac{15}{2})

Multiplying by (12) gives (9-6x+4x-24>0), so (x<-\frac{15}{2}). Reverse the sign when a negative coefficient remains.

Correct answer: D. (x<12)

Multiplying by (8) gives (5x+4<4x-8+24), so (x<12). Combine constants correctly after clearing denominators.

Correct answer: A. (\varnothing)

Subtracting (2x) from both sides gives (5>9), which is false. In such a case, there is no solution.

Correct answer: B. (x\in \mathbb{R})

Subtracting (3x) from both sides gives (-4\le 1), which is always true. Such an inequality has all real numbers as solution.

Correct answer: D. (-2<x\le 4)

Solve each inequality separately and then take the values common to both results. From \\(-3x+5\ge -7\\), subtract 5 to get \\(-3x\ge -12\\). Dividing by the negative number \\(-3\\) reverses the inequality sign, giving \\(x\le 4\\). From \\(x+2>0\\), subtract 2 to obtain \\(x>-2\\). Both conditions must hold at the same time, so the solution is \\(-2<x\le4\\).

Option D states exactly this common solution set. The endpoint \\(-2\\) is excluded because the second inequality is strict: at \\(x=-2\\), it gives 0>0, which is false. The endpoint 4 is included because the first inequality allows equality, and the second inequality is also satisfied there. Reversing the sign while dividing by \\-3 would incorrectly produce another option.

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

(3x+7\ge -8) gives (x\ge -5), so the interval is ([-5,\infty)). Use a closed endpoint for (\ge).

Correct answer: B. \(( 6, \infty )\)

Since 7 is positive, multiplying both sides by 7 does not change the inequality sign: \(5-2x<-7\). Thus, \(-2x<-12\). Dividing by \(-2\) reverses the inequality, giving \(x>6\). Hence, the interval is \((6,\infty)\). Option C incorrectly includes 6, whereas the original inequality is strict. Exam tip: Always reverse an inequality when multiplying or dividing by a negative number.

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

Adding 3 to both sides of \(2x-3\le 5\) gives \(2x\le 8\). Dividing by 2 gives \(x\le 4\). Therefore, \(x=5\) is greater than 4 and does not satisfy the inequality. The closest option, \(x=4\), does satisfy it because the equality case is included. Exam tip: In an inequality containing \(\le\), the boundary value is also a solution.

Correct answer: D. 5

Subtracting 1 from all three parts gives \(-6<3x\le 9\). Dividing by 3 gives \(-2<x\le 3\). The integers in this range are \(-1,0,1,2,3\), so there are 5 solutions. Option 4 misses \(x=3\), which is included because the upper inequality is \(\le\). Exam tip: In a compound inequality, check strict and inclusive endpoints separately.

Correct answer: B. (x>\frac{11}{9})

Multiplying by (6) gives (9x-10>1), so (x>\frac{11}{9}). Take the LCM of all fractions together.

Correct answer: C. (x\le \frac{18}{5})

Multiplying by (6) gives (4x-14\le 4-x), so (x\le \frac{18}{5}). Combine variable terms from both sides carefully.

Correct answer: D. \(x>-15\)

Given \(\frac{x+9}{-3}<2\), multiply both sides by \(-3\). Since the multiplier is negative, the inequality sign reverses: \(x+9>-6\). Therefore, \(x>-15\). Option B incorrectly includes \(x=-15\), which is not allowed because the original inequality is strict. Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative number.

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

Multiplying by (5) gives (-(2x-1)\ge 15), so (x\le -7). Apply the negative sign to the whole numerator.

Correct answer: B. (x\ge 17)

Multiplying by (2) gives (16-(x-3)\le 2), so (x\ge 17). Change signs properly when opening brackets after subtraction.

Correct answer: C. (x<\frac{9}{2})

Multiplying by (4) gives (12-(2x+1)>2), so (x<\frac{9}{2}). Remember sign reversal with a negative variable coefficient.

Correct answer: D. (x\ge 1)

Multiplying by (6) gives (2(5x-2)-(2x+1)\ge 3x), so (x\ge 1). Put brackets around the subtracted numerator.

Correct answer: A. (8)

(4x\le 32) gives (x\le 8), so the maximum value is (8). For maximum value, check the upper bound.