Algebraic Solution of Linear Inequalities in One Variable

Correct answer: A. (-3\leq x<4)

Subtracting (2) from all parts gives (-6\leq2x<8) and then (-3\leq x<4). Keep open and closed signs correctly.

Correct answer: A. 5 < x ≤ 14

The governing concept is transforming a compound inequality while preserving strict and non-strict endpoints. Begin with 1 < (x − 2)/3 ≤ 4. Multiply all three parts by 3. Because 3 is positive, the signs remain unchanged: 3 < x − 2 ≤ 12. Add 2 throughout to get 5 < x ≤ 14. Thus option A is correct. Notice that 5 is excluded because the first original sign was <, whereas 14 is included because the second original sign was ≤. Option B reverses which endpoint is included, option C omits all values between 5 and 14, and option D gives the wrong direction and range. In interval notation the answer is (5, 14], which clearly reflects the open left endpoint and closed right endpoint.

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

To change an inequality into interval form, first isolate the variable and then show whether the endpoint is included. Starting with 4x+1≥9, subtracting 1 from both sides gives 4x≥8. Dividing by the positive number 4 does not reverse the inequality, so x≥2. Thus, all real numbers from 2 onward are included.

The interval for x≥2 is [2,∞). The square bracket at 2 shows that 2 itself is allowed because equality is included. Infinity always uses a parenthesis because it is not an actual number that can be included as an endpoint. Therefore, option A is correct. Option B would represent x>2 and would incorrectly exclude 2.

Correct answer: A. ({x:x\in\mathbb{R},x\leq-1})

The inequality (x\leq-1) includes (-1) and all smaller real numbers. For real solutions writing (\mathbb{R}) is appropriate.

Correct answer: A. ({x:x\in\mathbb{R},x\geq3})

The inequality is 9−3x≤0. The important point is that the coefficient of x is negative. Subtracting 9 from both sides gives −3x≤−9. When both sides are divided by −3, the inequality sign must reverse because division by a negative number reverses order. This gives x≥3.

The solution set therefore contains 3 and every real number greater than 3. In set-builder form it is {x:x∈ℝ, x≥3}, which is exactly option A. The value x=3 checks the boundary because 9−3(3)=0, and 0≤0 is true. A choice using x≤3 would fail for a value such as x=4, so reversing the sign is essential.

Correct answer: A. 4

Solving the inequality gives \(3x-4\ge 8\Rightarrow 3x\ge 12\Rightarrow x\ge 4\). Since \(x\) is an integer, the smallest possible value is 4. Option 3 is incorrect because \(3(3)-4=5\), which is less than 8. Exam tip: dividing both sides of an inequality by a positive number does not reverse the inequality sign.

Correct answer: A. 5

Solving the inequality, \(5-2x>-7\) gives \(-2x>-12\). Dividing both sides by \(-2\) reverses the inequality sign, so \(x<6\). Since \(x\) is an integer, the greatest integer less than 6 is 5. Exam tip: Always reverse the inequality sign when multiplying or dividing by a negative number.

Correct answer: A. (x=2)

The solution is (x<3) so (x=2) satisfies it. Finding the general solution first is faster than testing all options.

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

Solving \(7-3x\leq 1\) gives \(-3x\leq -6\). Dividing both sides by \(-3\) reverses the inequality sign, so \(x\geq 2\). Therefore, \(x=2,3,4\) are solutions, whereas \(x=1\) is not. Exam tip: You can also substitute each option directly into the original inequality for a quick check.

Correct answer: A. The inequality sign reverses

Dividing by a negative number reverses the order of an inequality. This is the most common mistake in linear inequalities.

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

In \(-5x<20\), dividing both sides by \(-5\) reverses the inequality sign because the divisor is negative. Thus, \(x>\frac{20}{-5}=-4\), so option A is correct. Option B incorrectly keeps the original sign, while options C and D use the wrong sign for \(20\div(-5)\). Exam tip: always reverse an inequality sign when multiplying or dividing by a negative number.

Correct answer: A. \(x\leq-5\)

Dividing both sides of \(-2x\geq10\) by \(-2\) reverses the inequality sign because the divisor is negative: \(x\leq-5\). The value \(-5\) is included because the original relation was \(\geq\). Option B repeats the mistake of not reversing the sign. Exam tip: Always reverse an inequality sign when multiplying or dividing by a negative number.

Correct answer: A. \(3x+2<14\)

In option A, \(3x+2<14\). Subtracting 2 from both sides gives \(3x<12\), and dividing by 3 gives \(x<4\). Option B gives \(x>4\), option C also gives \(x>4\), while option D gives \(x\geq4\). Therefore, only option A has the stated solution set. Exam tip: Dividing an inequality by a positive number does not reverse its sign.

Correct answer: A. \(4x+8\ge 0\)

In option A, \(4x+8\ge 0\) gives \(4x\ge -8\), and dividing by 4 gives \(x\ge -2\). Since 4 is positive, the inequality sign remains unchanged. Option B gives \(x\le -2\), option C gives \(x<-2\), and option D gives \(x\ge 2\). Exam tip: dividing or multiplying an inequality by a positive number does not reverse its sign.

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

The statement translates to the inequality \(x+6<20\). Subtracting 6 from both sides gives \(x<14\), so option A is correct. Option C is incorrect because when \(x=14\), the result is exactly 20, not less than 20. Exam tip: ‘less than’ indicates the strict inequality \(<\), whereas ‘less than or equal to’ indicates \(\leq\).

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

“Not more than 28” means less than or equal to 28, so the inequality is \(4x\leq28\). Dividing both sides by the positive number 4 does not reverse the inequality sign, giving \(x\leq7\). Option B is incorrect because it excludes \(x=7\), even though equality is allowed. Exam tip: “not more than” means \(\leq\).

Correct answer: A. (a>18)

More than means strictly greater so (a>18). If the age is exactly (18), the condition is not satisfied.

Correct answer: A. 3

Solving \(3x+2\leq 11\) gives \(3x\leq 9\), and hence \(x\leq 3\). Therefore, the greatest integer satisfying the inequality is \(3\). The nearby distractor 4 is incorrect because it gives \(3(4)+2=14>11\); 5 also fails. Exam tip: dividing an inequality by a positive number does not reverse its sign.

Correct answer: A. 4

Starting with \(5x-1>14\), add 1 to both sides to get \(5x>15\), and then divide by 5 to obtain \(x>3\). Therefore, the least natural number greater than 3 is 4. Since the inequality is strict, \(x=3\) is not included.

Correct answer: A. \(0<k\leq 4\)

Substituting \(x=2\) into \(kx+1\leq 9\) gives \(2k+1\leq 9\). Thus, \(2k\leq 8\), so \(k\leq 4\). Since the question also states \(k>0\), combining both conditions gives \(0<k\leq 4\). Option B is incorrect because values \(k>4\) do not satisfy the inequality. In such questions, first substitute the given value and then take the intersection of all conditions.

Correct answer: A. 9

Solving \(x+2<k\) for \(x\) gives \(x<k-2\). Comparing this with the given solution \(x<7\), we get \(k-2=7\), so \(k=9\). The value 7 is the boundary of the solution set, not the value of \(k\). Exam tip: first isolate the variable and then compare the resulting boundary values.

Correct answer: A. 2x + 3 < 2x + 5

The governing concept is solving an inequality by simplifying both sides and identifying whether the variable remains. For option A, subtracting 2x from both sides gives 3 < 5. This statement is true independently of x, so every real number satisfies the original inequality. In option B, cancellation gives 5 < 3, which is false for every x. In option C, subtracting x gives 1 < -1, also always false. In option D, subtracting 3x gives 0 > 1, which is false. Therefore only option A represents an inequality with the solution set all real numbers. The result is an identity-type inequality rather than a condition restricting x.

Correct answer: A. \(x\geq4\)

Expanding the brackets gives \(2x+6-x\geq10\), which simplifies to \(x+6\geq10\). Subtracting 6 from both sides yields \(x\geq4\), so option A is correct. Option B has the inequality direction reversed. Remember that the inequality sign reverses only when multiplying or dividing by a negative number.

Correct answer: A. (x<20)

Multiplying by (6) gives (3(x+4)-2(x-2)<30). Simplifying gives (x+16<30), so the correct result is (x<14).

Correct answer: A. (x\geq\frac{13}{2})

The direct answer is A: x≥13/2. Start with (2x−5)/4+1≥3. Subtract 1 from both sides: (2x−5)/4≥2. Multiply both sides by 4. Since 4 is positive, the inequality sign stays the same: 2x−5≥8. Add 5: 2x≥13. Divide by 2, again positive, so x≥13/2. Option A is correct and includes equality because the original sign is ≥. Option B, x≤13/2, reverses the direction without a valid reason; division was by positive 2. Option C, x≥7/2, comes from an arithmetic error, usually forgetting that 5 must be added after multiplying. Option D, x≤7/2, has both the wrong boundary and wrong direction. A quick check with x=13/2 gives left side 3, so equality is accepted. A value smaller than 13/2, such as x=6, gives a value below 3 and fails. Memory cue: reverse an inequality only when multiplying or dividing by a negative number.