Algebraic Solution of Linear Inequalities in One Variable

Multiplying both sides by the positive number 10 gives \(8x+6>20-(x-1)=21-x\). Hence, \(9x>15\), so \(x>\frac{5}{3}\). The option \(x<\frac{5}{3}\) reverses the inequality incorrectly; multiplication by a positive number does not change the inequality sign. Exam tip: while removing brackets, remember that \(-(x-1)=-x+1\).

Given \(2(3-x)>x+9\), expand the bracket: \(6-2x>x+9\). Subtracting \(x\) from both sides gives \(6-3x>9\), so \(-3x>3\). On dividing by \(-3\), the inequality sign reverses; hence \(x<-1\). The option \(x\leq -1\) is incorrect because the original inequality is strict, so \(x=-1\) is not included. Exam tip: always reverse the inequality sign when multiplying or dividing by a negative number.

Multiplying both sides by the positive number 4 does not change the inequality sign: \(7-3x<2x+2\). Hence \(5<5x\), so \(x>1\). Option \(x\geq1\) is not correct because at \(x=1\), 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.

Since 3 is positive, multiplying every part of the compound inequality by 3 does not reverse the inequality signs: \(-12\leq x-1<6\). Adding 1 to all three parts gives \(-11\leq x<7\). Hence, option A is correct. Option B incorrectly stops before adding 1 to isolate \(x\). Exam tip: In a compound inequality, apply each operation to the left, middle, and right parts alike.

The first inequality gives (x\geq -5), and the second gives (x<5). Integers from (-5) to (4) are (10) in total.

Multiplying by (14) gives (42-4x+10\leq 7x+28). Thus (24\leq 11x) and (x\geq \frac{24}{11}).

From \(kx-3\leq 9\), we get \(kx\leq 12\). Since \(k>0\), dividing by \(k\) does not reverse the inequality, so \(x\leq \frac{12}{k}\). The given solution is \(x\leq 4\); hence \(\frac{12}{k}=4\), giving \(k=3\). If \(k=4\), the boundary would be \(x\leq 3\), so it is not correct. Exam tip: dividing an inequality by a positive number keeps its sign unchanged, while dividing by a negative number reverses it.

Multiplying both sides by the positive number 10 gives \(2(1-2x)\leq 5(3-x)\), or \(2-4x\leq 15-5x\). Adding \(5x\) to both sides gives \(2+x\leq 15\), hence \(x\leq 13\). The closest distractor is \(x<13\), but \(x=13\) makes both sides equal, so it must be included. Exam tip: multiplying an inequality by a positive number does not reverse its sign.

Solving the inequality, \(4x-7\leq 2x+1\) gives \(2x\leq 8\), hence \(x\leq 4\). Since \(x\) is a natural number, the greatest natural number satisfying the condition is \(4\). Option 5 is incorrect because substituting \(x=5\) does not satisfy the inequality. Exam tip: Bring variable terms to one side and constants to the other before checking the boundary value.

Simplifying gives \(5x-2(3x-4)\geq11\Rightarrow5x-6x+8\geq11\Rightarrow-x\geq3\). On multiplying both sides by \(-1\), the inequality sign reverses, so \(x\leq-3\). The option \(x<-3\) is incorrect because \(x=-3\) also satisfies the equality. Exam tip: always reverse the inequality sign when multiplying or dividing by a negative number.

Multiplying both sides of the inequality by 10 gives \(5x+30>4x-2\). Subtracting \(4x\) from both sides gives \(x+30>-2\), so \(x>-32\). The option \(x\geq -32\) is incorrect because at \(x=-32\), both sides are equal, whereas the given sign is \(>\). Exam tip: multiplying or dividing an inequality by a positive number does not reverse its sign.

Multiplying both sides of the inequality by the positive number \(8\) gives \(3x+2<4(x-1)+8\). Simplifying, \(3x+2<4x+4\), so \(-2<x\), i.e. \(x>-2\). The option \(x\geq -2\) is incorrect because the original inequality is strict, so \(x=-2\) is not included. Exam tip: The inequality sign does not reverse when multiplying by a positive number.

Simplifying the left-hand side gives \(12-3(2x+1)=12-6x-3=9-6x\). Thus, \(9-6x\leq x-5\), which gives \(14\leq 7x\). Dividing by 7 gives \(x\geq 2\), so option A is correct. The option \(x>2\) incorrectly excludes \(x=2\), which satisfies the inequality. Exam tip: the inequality sign does not change when dividing by a positive number.

Multiplying both sides of \(\frac{x-6}{-3}>2\) by \(-3\) reverses the inequality sign: \(x-6<-6\). Therefore, \(x<0\). The option \(x>0\) results from forgetting to reverse the sign when multiplying by a negative number. Exam tip: Always reverse an inequality sign when multiplying or dividing by a negative quantity.

Multiplying both sides of the inequality by the positive number 6 gives \(4x+10\leq 42-x\). Hence, \(5x\leq 32\), so \(x\leq \frac{32}{5}\). The option \(x<\frac{32}{5}\) is incorrect because equality holds at \(x=\frac{32}{5}\). Exam tip: Multiplying an inequality by a positive number does not reverse its sign.

From \(2x-3<5\), we get \(2x<8\), so \(x<4\). From \(x+4\geq1\), we get \(x\geq-3\). Satisfying both conditions gives \( -3 \leq x < 4 \), so option A is correct. Option B incorrectly excludes \(-3\) and includes \(4\). Exam tip: the word “and” means take the intersection of the solution sets.

Expanding the left-hand side gives \(6x-3-4x-8>x-9\), or \(2x-11>x-9\). Adding \(x\) to both sides gives \(x>2\). \(x\geq2\) is incorrect because the original inequality is strict, so \(x=2\) is not included. Exam tip: Reverse the inequality sign only when multiplying or dividing by a negative number.

Multiplying both sides by the positive number \(6\) does not reverse the inequality: \(5x-4\geq 2(x+2)+6\). Hence \(5x-4\geq 2x+10\), so \(3x\geq 14\) and \(x\geq \frac{14}{3}\). The option \(x>\frac{14}{3}\) is incorrect because \(x=\frac{14}{3}\) also satisfies the inequality. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.

From \(mx+2>8\), we get \(mx>6\). Since the solution is \(x<-3\), \(m\) must be negative because dividing an inequality by a negative number reverses its sign. Hence the boundary value satisfies \(\frac{6}{m}=-3\), which gives \(m=-2\). If \(m=2\), the solution would be \(x>3\), so it is the closest incorrect option. Exam tip: always reverse the inequality sign when multiplying or dividing by a negative number.

Multiplying both sides by the positive number 8 does not change the inequality sign: \(2(x+1)+(x+3)>8\). Hence, \(3x+5>8\), so \(3x>3\) and \(x>1\). The option \(x\geq1\) is incorrect because at \(x=1\), the left-hand side equals 1, not a value greater than 1. Exam tip: multiplying an inequality by a positive number keeps its sign unchanged.

When both sides of an inequality are multiplied or divided by a negative number, the inequality sign reverses. Therefore, multiplying \(p<q\) by \(-5\) gives \(-5p>-5q\). Adding the same number to both sides or dividing by a positive number does not change the sign. Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative quantity.

Multiply both sides by the positive number 12, so the inequality sign remains unchanged: \(24-3(3x-1)<4(x+2)\). This gives \(27-9x<4x+8\), hence \(19<13x\) and \(x>\frac{19}{13}\). The option \(x<\frac{19}{13}\) reverses the inequality sign incorrectly. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.

From the first inequality, \(5-2x\geq -1\) gives \(-2x\geq -6\). Dividing by \(-2\) reverses the inequality, so \(x\leq 3\). From \(3x+4>1\), we get \(3x>-3\), hence \(x>-1\). Satisfying both conditions simultaneously gives \(-1<x\leq3\). Option B has the endpoint signs reversed. Exam tip: always reverse an inequality sign when multiplying or dividing by a negative number.

Multiplying both sides by the positive number \(4\) does not change the inequality sign: \(2(x-5)\leq 3-x\). Thus, \(2x-10\leq 3-x\), so \(3x\leq 13\), giving \(x\leq \frac{13}{3}\). Option C incorrectly excludes \(x=\frac{13}{3}\), which satisfies the inequality. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.

Multiplying both sides by 6 does not reverse the inequality because 6 is positive: \(9x-24<2x+18\). Hence \(7x<42\), so \(x<6\). The option \(x\leq6\) is incorrect because the original inequality is strict; at \(x=6\), both sides are equal. Exam tip: reverse an inequality sign only when multiplying or dividing by a negative number.