Introduction to inequalities

Correct answer: A. adding (7) to both sides

Adding the same number to both sides does not change inequality direction. The sign changes only when multiplying or dividing by a negative number.

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

We get (-4x\ge 12), and dividing by (-4) reverses the sign to (x\le -3). A negative coefficient is the most common trap.

Correct answer: C. (x<-1) and (x>3)

The same (x) cannot be less than (-1) and greater than (3) at once. For compound inequalities, find the common part.

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

The inequality (x\le -2) includes (-2) and all smaller real values. Infinity always uses an open parenthesis.

Correct answer: C. (m=n)

Being less than or equal in both directions means equality. Recognizing this helps in comparison questions.

Correct answer: A. (3x+4>10)

The direct answer is A: \(3x+4>10\). Start with \(x>2\). Multiplying both sides by the positive number 3 keeps the inequality direction: \(3x>6\). Adding 4 to both sides gives \(3x+4>10\). The inequality is strict because \(x\) is strictly greater than 2; it cannot equal 2. Option A is correct. Option B, \(3x+4\ge10\), is weaker and is not the exact conclusion: equality would require \(x=2\), which is forbidden. Option C reverses the relationship and is false; for example, if \(x=3\), the expression is 13, not less than 10. Option D says the expression always equals 10, but its value changes with \(x\). Remember: multiplying or dividing by a positive number keeps the sign; adding or subtracting also keeps it. Only multiplication or division by a negative number reverses an inequality.

Correct answer: B. (-5x+2\ge 7)

Multiplying (x\le -1) by (-5) gives (-5x\ge 5), then adding (2) gives (-5x+2\ge 7). Negative multiplication reverses the sign.

Correct answer: A. (x\ge 11)

Multiplying both sides by positive (2) gives (x-3\ge 8), so (x\ge 11). A positive denominator does not change the sign.

Correct answer: B. (x> -8)

Multiplying by (-3) reverses the sign, so (5-2x>-21), then (-2x>-26), and dividing by (-2) gives (x<13). The listed choices do not match this result.

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

Multiplying by (-2) gives (7-3x\ge -10), then (-3x\ge -17), so (x\le \frac{17}{3}). The correct solution is not among the options, so this question is invalid.

Correct answer: D. no definite conclusion

Inequalities in opposite directions do not give a fixed order when added directly. The directions should match before adding.

Correct answer: A. ([0,5))

The direct answer is A: \([0,5)\). The interval \(x\in[2,7)\) means \(2\le x<7\). Subtract 2 from every part: \(2-2\le x-2<7-2\), so \(0\le x-2<5\). Therefore the answer is \([0,5)\). The left endpoint is closed because \(x=2\) is allowed, and then \(x-2=0\). The right endpoint is open because \(x=7\) is not allowed, so 5 cannot be included. Option A has exactly these endpoints. Option B, \((0,5]\), wrongly excludes 0 and includes 5. Option C, \([0,5]\), wrongly includes 5. Option D, \((0,5)\), wrongly excludes both valid endpoint 0 and the rest of the correct boundary information. A translation of an interval changes its numerical endpoints but does not change an open bracket into a closed bracket. Memory cue: subtract the same number from both ends, then copy the bracket types.

Correct answer: C. \(x>-1\) and \(x\le 6\)

The compound inequality \(-1<x\le 6\) means two conditions: (i) \(x>-1\) and (ii) \(x\le 6\). Both must be true at the same time, so the equivalent statement is \(x>-1\) and \(x\le 6\).

Correct answer: B. \(( -\infty,-3]\cup[3,\infty)\)

For \(|x|\ge 3\), we have \(x\le -3\) or \(x\ge 3\). Converting to intervals gives \(( -\infty,-3]\cup[3,\infty)\).

Correct answer: A. (x^2>1)

When (x>1), (x) is positive and greater than (1), so its square is also greater than (1). Check the sign condition before squaring.

Correct answer: B. \(x\ge -2\)

The negation of \(x<-2\) includes every value that is not less than \(-2\), so it is \(x\ge -2\). Equality must be included when negating a strict inequality. Option C is incorrect because it includes values less than or equal to \(-2\), rather than the values outside \(x<-2\). Exam tip: the negation of \(x<a\) is \(x\ge a\), while the negation of \(x>a\) is \(x\le a\).

Correct answer: A. (x<z)

From (x\le y) and (y<z), we get (x<z) because one link is strict. In a mixed chain, a strict sign can make the result strict.

Correct answer: A. (ka<kb)

Multiplying by a positive number does not change inequality direction. Check the sign of the multiplier before writing the conclusion.

Correct answer: A. (-10<2-3x\le 11)

Multiplying (-3\le x<4) by (-3) gives (-12<-3x\le 9), then adding (2) gives (-10<2-3x\le 11). Negative multiplication reverses the order.

Correct answer: A. (x=\frac{1}{2})

At (x=\frac{1}{2}), (x^2=\frac{1}{4}), which is less than (\frac{1}{2}). One counterexample is enough to disprove a universal statement.

Correct answer: A. The inequality is true for all real values of \(x\).

\(x^2\ge 0\) for all real \(x\). So \(x^2+1\ge 1>0\). Hence \(x^2+1>0\) is always true, for every real \(x\).

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

When (x>1), (\frac{1}{x}<1), so (x>\frac{1}{x}). Positivity is important in reciprocal comparisons.

Correct answer: A. both assertion and reason are true, and the reason explains it

The direct answer is A: both the assertion and reason are true, and the reason correctly explains the assertion. Start with a<b. Adding the same number c to both sides gives a+c<b+c. This operation shifts both numbers equally on the number line, so their order does not change. No restriction is needed on c; it may be positive, zero, or negative. A is correct because the reason states exactly the rule used in the proof. B is wrong because the reason is not merely true separately—it directly justifies the assertion. C is wrong because the reason is true, not false. D is wrong because the assertion is true, not false. Do not confuse addition with multiplying or dividing by a negative number: adding the same quantity preserves the inequality direction. Exam cue: add or subtract the same number on both sides, and the sign stays unchanged.

Correct answer: A. (2x-5>1)

Multiplying (x>3) by (2) gives (2x>6), and subtracting (5) gives (2x-5>1). Positive multiplication does not reverse the sign.

Correct answer: A. \(a<c\)

By the transitive property of inequalities, \(a\le b\) and \(b<c\) imply \(a<c\). Even when \(a=b\), we have \(a=b<c\), so the final inequality remains strict. Equality in option B is not necessary, while options C and D contradict the given chain. Exam tip: in a chain of mixed inequalities, one strict comparison is enough to make the comparison between the outer terms strict.