Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. (x>\frac{1}{20})

Multiplying by (18) gives (3(4x-5)+2(x+2)<9(3x+1)), so (x>\frac{1}{20}). After clearing denominators, simplify both sides carefully.

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

Given \(3x-7<11\), add 7 to both sides to get \(3x<18\). Dividing both sides by the positive number 3 does not reverse the inequality sign, so \(x<6\). The option \(x\leq 6\) is incorrect because equality is not included in the original inequality. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: B. \(x\le -6\)

Subtracting 5 from both sides gives \(-2x\ge 12\). On dividing by \(-2\), the inequality sign reverses, so \(x\le -6\). Option C is a close distractor, but \(x=-6\) is included because the original inequality has \(\ge\). Exam tip: always reverse the inequality sign when multiplying or dividing by a negative number.

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

Multiplying both sides of \(\frac{x-3}{4}+2>\frac{x}{2}\) by 4 gives \(x-3+8>2x\). Thus, \(x+5>2x\), which gives \(5>x\), or \(x<5\). \(x\leq 5\) is not correct because the original inequality is strict; at \(x=5\), both sides are equal. Exam tip: Multiplying an inequality by a positive number does not reverse its sign.

Correct answer: D. \(x\le 6\)

Adding \(1.2\) to both sides gives \(0.3x\le 1.8\). Dividing both sides by the positive number \(0.3\) does not reverse the inequality sign, so \(x\le 6\). The close distractor \(x<6\) is incorrect because \(x=6\) also satisfies the original inequality. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. \(x > -\frac{4}{5}\)

Given \(-4(2x-1)<3(1-x)+5\), expanding the brackets gives \(-8x+4<8-3x\). Hence, \(-4<5x\), so \(x>-\frac{4}{5}\). Option B has the inequality in the wrong direction, while option C incorrectly includes equality; the original inequality is strict. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

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

Given \(7-(3x+2)\ge 2x-10\), opening the bracket gives \(7-3x-2\ge 2x-10\), or \(5-3x\ge 2x-10\). Adding \(3x\) and \(10\) to both sides gives \(15\ge 5x\). Since 5 is positive, dividing by 5 does not reverse the inequality sign, so \(x\le 3\). The closest distractor, \(x<3\), is incorrect because \(x=3\) also satisfies the inequality. Exam tip: reverse an inequality only when multiplying or dividing by a negative number.

Correct answer: A. \(x\le -14\)

Multiplying both sides by the positive number 6 does not change the inequality sign: \(2(2x+1)\le 3(x-4)\), so \(4x+2\le 3x-12\). Subtracting \(3x\) and 2 from both sides gives \(x\le -14\). Hence, option A is correct. The closest distractor, \(x<-14\), excludes \(x=-14\), although this value satisfies the inequality. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

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

Simplify the left-hand side: \(2(3x-5)-4(x+1)=6x-10-4x-4=2x-14\). Thus, \(2x-14>8\), so \(2x>22\), giving \(x>11\). \(x\geq11\) is not correct because the original inequality is strict; at \(x=11\), both sides become equal. Exam tip: while expanding \(-4(x+1)\), apply the negative factor to both terms.

Correct answer: A. \(x<\frac{12}{7}\)

Multiplying both sides by the positive number \(14\) does not change the inequality sign: \(2(5x-2)<3x+8\). Thus, \(10x-4<3x+8\), so \(7x<12\) and hence \(x<\frac{12}{7}\). The option \(x\leq\frac{12}{7}\) is incorrect because at \(x=\frac{12}{7}\), both sides are equal, whereas the question has a strict \(<\) inequality. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: B. \(x\ge \frac{75}{11}\)

Multiplying both sides by 15 gives \(-5x+60\le 6x-15\). Moving 60 and \(-5x\) to the other sides gives \(75\le 11x\). Since 11 is positive, dividing by it does not reverse the inequality sign, so \(x\ge \frac{75}{11}\). Option C excludes equality, but the original inequality uses \(\le\). Exam tip: An inequality sign changes direction only when multiplying or dividing by a negative number.

Correct answer: A. \(x \ge \frac{13}{3}\)

Simplifying the right-hand side gives \(2(1-x)+15=17-2x\). Thus, \(4x-9\ge 17-2x\), so \(6x\ge 26\) and hence \(x\ge \frac{13}{3}\). The option \(x>\frac{13}{3}\) is incorrect because \(x=\frac{13}{3}\) also satisfies the inequality. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. \(x\ge 17\)

Using the common denominator 4, \(\frac{3x+5}{4}-\frac{x-1}{2}=\frac{3x+5-2(x-1)}{4}=\frac{x+7}{4}\). Hence, \(\frac{x+7}{4}\ge 6\Rightarrow x+7\ge24\Rightarrow x\ge17\). Option \(x>17\) is the closest distractor, but \(x=17\) also satisfies the inequality. Exam tip: multiplying an inequality by a positive number does not reverse its sign.

Correct answer: A. \(x<0\)

Multiplying both sides by the positive number 6 does not change the inequality sign: \(4-x>2(x+2)\). Thus, \(4-x>2x+4\), giving \(-3x>0\). On dividing by the negative number \(-3\), the inequality sign reverses, so \(x<0\). The closest distractor, \(x\leq0\), is incorrect because at \(x=0\) both sides are equal, whereas the question uses \(>\). Exam tip: always reverse an inequality when multiplying or dividing by a negative number.

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

Subtracting \(0.6x\) and \(2.4\) from both sides gives \(0.9x<-5.4\). Since \(0.9\) is positive, dividing both sides by \(0.9\) does not reverse the inequality sign, so \(x<-6\). The option \(x\le -6\) is incorrect because the original inequality is strict and does not include equality. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. \(x \ge 1\)

Multiply both sides by the positive number 10 to remove the fractions: \(20x-6\ge 5x+9\). Hence, \(15x\ge 15\), so \(x\ge 1\). The closest distractor, \(x>1\), is incorrect because at \(x=1\) both sides are equal, so 1 is included in the solution. Exam tip: multiplying an inequality by a positive number does not reverse its sign.

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

On expanding the brackets, \(6-5(x-2)=16-5x\) and \(3(2-x)=6-3x\). Therefore, \(16-5x\le 6-3x\), which gives \(10\le 2x\). Since 2 is positive, dividing by 2 does not reverse the inequality sign, so \(x\ge 5\). The option \(x\le 5\) is incorrect because an inequality reverses only when multiplying or dividing by a negative number. Exam tip: after expanding brackets, move like terms carefully and check whether any operation involves a negative number.

Correct answer: B. No solution

The left side is (5x-10), and (5x-10\le 5x-7) is always true. Hence the answer should be all real numbers.

Correct answer: C. No solution

Expanding the left-hand side gives \(4(2x+3)=8x+12\). Thus, the inequality becomes \(8x+12>8x+15\). Subtracting \(8x\) from both sides gives \(12>15\), which is false. Hence, the inequality is not true for any real value of \(x\), so there is no solution. Options such as \(x>3\) or \(x<3\) are incorrect because the variable terms cancel completely. Exam tip: When the coefficients of the variable are the same on both sides, cancel those terms first and compare the constants.

Correct answer: A. \(x \ge \frac{13}{7}\)

Multiplying both sides by the positive number 10 does not change the inequality sign: \(14-4x \le 3x+1\). Hence, \(13 \le 7x\), so \(x \ge \frac{13}{7}\). The option \(x>\frac{13}{7}\) is incorrect because at \(x=\frac{13}{7}\), both sides are equal, and \(\le\) includes equality. Exam tip: Reverse an inequality sign only when multiplying or dividing by a negative number.

Correct answer: A. \(x\le -2\)

Simplifying the left side gives \(-3(x+2)+7=-3x+1\), and simplifying the right side gives \(2(4-x)-5=3-2x\). Thus, \(-3x+1\ge 3-2x\), which gives \(-x\ge 2\), and hence \(x\le -2\). The option \(x<-2\) is close, but at \(x=-2\) both sides are equal, so \(-2\) must be included. Exam tip: reverse the inequality sign whenever you multiply or divide an inequality by a negative number.

Correct answer: A. \(x>2\)

Write the second fraction with denominator 8: \(\frac{x+6}{8}-\frac{2(x-2)}{8}=\frac{10-x}{8}\). Therefore, \(\frac{10-x}{8}<1\Rightarrow 10-x<8\Rightarrow -x<-2\). Multiplying by \(-1\) reverses the inequality sign, so \(x>2\). The option \(x\geq2\) is incorrect because the original inequality is strict; at \(x=2\), the left-hand side equals 1. Exam tip: always reverse an inequality sign when multiplying or dividing by a negative number.

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

Given \(11-3x>2x+1\), subtracting \(1\) from both sides gives \(10-3x>2x\). Adding \(3x\) to both sides gives \(10>5x\), so \(x<2\). Hence, the interval-form solution is \(( -\infty, 2)\). Option C incorrectly includes \(2\); a strict inequality \(>\) does not include the endpoint. Exam tip: use an open bracket at an endpoint for strict inequalities \((<,>)\).

Correct answer: A. \(x>-\frac{14}{5}\)

Subtracting \(18\) from both sides and adding \(7x\) gives \(-5x<14\). On dividing by \(-5\), the inequality sign reverses, so \(x>-\frac{14}{5}\). The option \(x\geq-\frac{14}{5}\) is incorrect because the original inequality is strict; at \(x=-\frac{14}{5}\), both sides become equal. Exam tip: always reverse the inequality sign when multiplying or dividing by a negative number.

Correct answer: A. \(x\ge 2\)

Expanding the left side gives \(2.5(2x-1)=5x-2.5\), while the right side gives \(1.5(x+3)=1.5x+4.5\). Hence, \(5x-2.5\ge 1.5x+4.5\), so \(3.5x\ge 7\). Since \(3.5\) is positive, dividing both sides by it does not reverse the inequality sign, giving \(x\ge 2\). The closest distractor is \(x>2\), but \(x=2\) makes both sides equal, so it must be included. Exam tip: An inequality sign changes direction only when multiplying or dividing by a negative number.