Algebraic Solution of Linear Inequalities in One Variable

Correct answer: C. (x<13)

Multiplying by (6) gives (3(x+3)-2(x-1)<24), hence (x<13). After clearing fractions, expand brackets carefully.

Correct answer: D. (x\ge \frac{9}{5})

Multiplying by (6) gives (2(2x-1)+(x+5)\ge 12), so (5x+3\ge 12) and (x\ge \frac{9}{5}). Use the LCM to clear denominators.

Correct answer: A. (x\ge 0)

Multiplying by (4) gives (5x+2\le 2(3x-1)+4), hence (x\ge 0). Multiply the entire right side while clearing denominators.

Correct answer: B. (x>28)

The correct answer is option B: x > 28. Start with 3x/4 − 2 > x/2 + 5. Move the variable terms to the left and constants to the right: 3x/4 − x/2 > 7. Since x/2 = 2x/4, the left side becomes x/4. Therefore x/4 > 7. Multiply by positive 4, so x > 28. Option B is exactly this result. Option A, x < 28, reverses the direction and would include values that do not work, such as x = 0. Option C, x > 7/4, comes from mishandling the coefficient and is much too weak; x = 2 would satisfy it but fail the original inequality. Option D, x ≤ 28, both reverses the direction and incorrectly includes equality. Equality is not allowed because the original sign is strictly greater than. Memory cue: simplify the coefficient difference first: 3/4 − 1/2 = 1/4, then multiply by 4.

Correct answer: C. (x\le -\frac{3}{8})

(10-5x\ge 13+3x) gives (-8x\ge 3), so (x\le -\frac{3}{8}). Reverse the sign when dividing by a negative coefficient.

Correct answer: A. (x\le -\frac{5}{7})

Multiplying by (12) gives (6x+6+4x+8\le 3x+9), so (x\le -\frac{5}{7}). Clear denominators and combine linear terms.

Correct answer: B. (x>\frac{16}{7})

Multiplying by (10) gives (8x-14>x+2), hence (x>\frac{16}{7}). Multiply the complete numerator while clearing fractions.

Correct answer: C. (x\ge 11)

Multiplying by (6) gives (4x-5\ge 3x+6), so (x\ge 11). Quickly identify the LCM of small denominators.

Correct answer: D. (x\ge \frac{13}{5})

Multiplying by (6) gives (14-4x\le x+1), hence (x\ge \frac{13}{5}). Moving the variable to the right can sometimes simplify the work.

Correct answer: A. \(5<x\le 14\)

Given \(1<\frac{x-2}{3}\le4\), multiply all three parts by 3. Since 3 is positive, the inequality signs do not change: \(3<x-2\le12\). Adding 2 throughout gives \(5<x\le14\). Hence, option A is correct. Option B incorrectly includes \(x=5\), although the left inequality in the original statement is strict. Exam tip: inequality signs reverse only when multiplying or dividing by a negative number.

Correct answer: B. (-\frac{11}{3}\le x<\frac{19}{3})

Multiplying by (5) gives (-10\le 3x+1<20), so (-\frac{11}{3}\le x<\frac{19}{3}). A positive multiplier does not change the signs.

Correct answer: A. (5)

(4x<21) gives (x<\frac{21}{4}), so the greatest integer is (5). When an integer answer is asked, check values near the boundary.

Correct answer: B. (3)

(-3x\le -9) gives (x\ge 3), so the least integer is (3). Check the boundary after division by a negative.

Correct answer: C. (6)

The direct answer is option C, 6. The inequality is \(-2<x\le4\). The open sign at -2 means -2 is excluded, while the closed sign at 4 means 4 is included. List the integers greater than -2 and at most 4: \(-1,0,1,2,3,4\). Counting them gives six integers. Option A, 5, misses one valid integer. Option B, 7, counts too many, usually by including -2 incorrectly or adding another number outside the interval. Option C, 6, matches the complete list. Option D, 4, omits two valid values. A number-line picture makes this clear: place an open circle at -2, a closed circle at 4, and count the integer marks between them. The common mistake is to treat an open endpoint as included. Here -2 must not be counted, but 4 must be counted.

Correct answer: D. (5)

(x<\frac{11}{2}), so the natural numbers are (1,2,3,4,5). For natural numbers, start counting from (1).

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

The direct answer is option B: \((-\infty,6]\). Solve first: \(2x-5\le7\). Add 5 to both sides to get \(2x\le12\). Divide by positive 2, giving \(x\le6\). In interval notation, all numbers extending left without bound are represented by \(-\infty\), and 6 is included because equality is allowed, so a square bracket is used at 6: \((-\infty,6]\). Option A uses a parenthesis and wrongly excludes 6. Option B is correct. Option C describes values from 6 upward, the opposite direction. Option D excludes 6 and goes rightward. Infinity always uses a parenthesis because it is not an actual endpoint.

Correct answer: C. (x>\frac{11}{2})

Multiplying by (12) gives (6x-9>4x+2), hence (x>\frac{11}{2}). Use the LCM to remove small fractions.

Correct answer: D. (x\le \frac{9}{7})

Multiplying by (4) gives (10-6x\ge x+1), so (x\le \frac{9}{7}). Combine variable terms before dividing.

Correct answer: A. 70

The condition is represented by \(x+5\geq 75\). Subtracting 5 from both sides gives \(x\geq 70\). Hence, the minimum original score is 70. Although 75 also satisfies the condition, it is not the minimum value. Exam tip: use \(\geq\) for the phrase “at least.”

Correct answer: B. (x<10)

The direct answer is option B: \(x<10\). “Less than 150” means a strict inequality, so write \(12x+30<150\). Subtract 30: \(12x<120\). Divide by positive 12: \(x<10\). Option A, \(x<12\), results from incorrect arithmetic or ignoring the constant term. Option B is correct and does not include 10; at \(x=10\), the cost is \(12(10)+30=150\), but the price must be less than 150. Option C uses \(\le\), which would mean “at most” and would allow 10. Option D reverses the direction and describes values greater than 10. In word problems, translate “less than” as < and “at most” as ≤.

Correct answer: A. \(-3<x\leq 7\)

The perimeter is \(2[(x+3)+5]\). Therefore, \(2x+16\leq30\), which gives \(2x\leq14\) and hence \(x\leq7\). Also, the length \(x+3\) must be positive, so \(x>-3\). Combining both conditions gives \(-3<x\leq7\). The option \(x\leq7\) satisfies only the perimeter condition and still includes values that give a non-positive length. Exam tip: after solving an inequality in a word problem, always check practical restrictions such as a length being positive.

Correct answer: D. (x>5)

The direct answer is option D: \(x>5\). The new temperature is \(40-3x\), and “less than 25” means \(40-3x<25\). Subtract 40 from both sides: \(-3x<-15\). Dividing by \(-3\) reverses the inequality sign, giving \(x>5\). Option A, \(x<5\), has the sign reversed and is therefore wrong. Option B, \(x\le5\), incorrectly includes 5, but at \(x=5\) the temperature is exactly 25, not less than 25. Option C, \(x>3\), does not follow from the algebra and is too weak; for example, \(x=4\) gives temperature 28. Option D is correct because every value greater than 5 makes the temperature below 25. Remember: when dividing an inequality by a negative number, always reverse its sign.

Correct answer: A. (x\ge 16)

Multiplying by (10) gives (2(2x+3)-5(x-4)\le 10), so (x\ge 16). Watch the signs after a subtraction before a bracket.

Correct answer: B. (x>\frac{31}{7})

Multiplying by (14) gives (2(3x-2)+(x+1)>28), hence (x>\frac{31}{7}). Use the same simplification process as equations while tracking the sign.

Correct answer: C. (x\le 1)

Multiplying by (12) gives (3x-15\ge 4(2x+1)-24), so (x\le 1). Combine constants carefully after clearing denominators.