Introduction to inequalities

The symbol < means “is less than.” Therefore, a < b means that the value of a is less than the value of b, so option A is correct.

The symbol \(\ge\) means ‘greater than or equal to’. Therefore, \(x\ge 7\) includes \(x=7\) as well as every number greater than 7. Option A is incorrect because it excludes \(x=7\). Exam tip: read \(\ge\) as ‘greater than or equal to’ and \(\le\) as ‘less than or equal to’.

A linear inequality in one variable has the variable only to the first power and does not contain products of variables, powers such as x², or non-linear operations such as a square root. In 2x+3<9, x has degree 1 and is multiplied only by the constant 2; the relation also uses an inequality sign, so it is a linear inequality. Option A is therefore correct. The expression x²<4 is quadratic because x has power 2. The expression √x<5 is not linear because it contains a radical, and xy>6 is not linear because it contains a product of variables. The classification concerns the algebraic form, not merely the presence of an inequality symbol.

Substituting \(x=3\) gives \(3<4\), which is true, so 3 is a solution. The closest distractor, 4, is not valid because \(4<4\) is false; the symbol '<' means strictly less than 4. In an exam, test each option by substituting its value into the inequality.

To test a value in an inequality, substitute that value directly into the statement. With x=2, the inequality x≤2 becomes 2≤2. The symbol ≤ means “less than or equal to,” so equality is included; a number equal to 2 satisfies the condition. Therefore the statement is true, and option A is correct. If the sign had been x<2, then x=2 would not satisfy it because strict inequality excludes equality. Option B confuses < with ≤. Option C is unnecessary because the value of x is explicitly given, and option D does not describe the logical truth value being asked. This illustrates the importance of reading the equality bar in an inequality sign.

“More than” means “greater than,” so the value of y must be greater than 10, giving y > 10. Option A represents y less than 10, option C represents equality, and option D represents y less than or equal to 10. Exam tip: translate “more than” or “greater than” as the symbol >.

“Not more than 12” means that p may be less than or equal to 12, so the correct inequality is \(p\le 12\). Option D, \(p<12\), is incorrect because it excludes the value p = 12. Exam tip: translate “not more than” and “at most” as \(\le\).

Adding the same number to both sides does not change the inequality direction. In addition or subtraction the sign stays the same.

The direct answer is option A, \\(4m<24\\). Begin with \\(m<6\\). Multiplying every part of an inequality by the positive number 4 gives \\(4m<4(6)\\), so \\(4m<24\\). The inequality sign stays in the same direction because 4 is positive. Option A is correct. Option B, \\(4m>24\\), incorrectly reverses the sign; reversal is required only for multiplication by a negative number. Option C, \\(m<24\\), changes the right side but does not multiply the left side, so it is not the result of multiplying both sides by 4. Option D, \\(m>24\\), has both an incorrect sign and incorrect transformation. For example, if m=5, then 5<6 and 20<24, confirming the rule. Positive multiplication keeps the sign; negative multiplication reverses it.

The governing rule is that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Starting with x>5, multiply each side by −1: (−1)x>(−1)5. Because −1 is negative, the greater-than sign changes to less-than, giving −x<−5. Therefore option B is correct. A is the common error of multiplying the numbers but leaving the sign unchanged. C changes x incorrectly and does not represent multiplication of the original left side by −1. D changes only the form of one side and fails to apply the operation consistently. A quick numerical check confirms the result: if x=6, then −6<−5 is true, whereas −6>−5 is false.

The symbol \(\ge\) means “greater than or equal to.” Thus, the solution includes the endpoint \(-2\) as well as every number larger than \(-2\). On a number line, numbers increase as we move to the right, so the solution extends from \(-2\) toward the right.

Because \(-2\) is included, it is shown with a closed or filled circle. An open circle would mean that the endpoint is excluded, as in a strict inequality such as \(x>-2\). Therefore the correct representation is a closed point at \(-2\) with shading or an arrow to the right, which is option B. The endpoint and direction must both be checked.

The symbol ≤ means less than or equal to. Therefore, the statement 3 ≤ 3 is true because 3 is equal to 3, and equality is explicitly included in the relation. Statement 5 < 2 is false because 5 is greater than 2. Statement 7 > 9 is false because 7 is less than 9. Statement 4 ≠ 4 is false because 4 is equal to itself, so it does not satisfy the not-equal sign. Only one option is true, namely option B. A useful method is to compare the numbers directly and pay careful attention to whether the symbol includes equality, as ≤ does, while < does not.

In (r\le 9), (9) and numbers smaller than (9) are included. With equality signs, the boundary value is also a solution.

“40 or more” means that the score may be exactly 40 or any value greater than 40, so the correct inequality is \(s\ge 40\). Option A is incorrect because \(s>40\) excludes a score of exactly 40. Exam tip: use \(\ge\) for “or more” and \(\le\) for “or less.”

The governing concept is translating an upper-limit statement into an inequality. The phrase “should not exceed 15 kilograms” means that the weight may be less than 15 kilograms or exactly 15 kilograms, but it cannot be greater than 15. Therefore the correct mathematical statement is w≤15. For example, 10 kg and 14.8 kg are allowed, and exactly 15 kg is also allowed; 15.1 kg is not allowed. Option A, w<15, wrongly excludes the permitted boundary value 15. Options C and D describe weights above 15 or at least 15, which express a lower limit rather than a maximum. The wording “not exceed” always includes the stated maximum when no additional restriction is given. Hence option B is the unique correct inequality.

“Less than” means strictly smaller than 100, so the correct inequality is \(p<100\). Option B is incorrect because \(p\le 100\) also includes \(p=100\), which is not allowed here. In exams, remember that “less than” uses \(<\), while “less than or equal to” uses \(\le\).

“Not less than 18” means “at least 18,” so the age can be 18 or any greater value: \(a\ge 18\). Option B, \(a\le 18\), means the age is at most 18 and reverses the intended condition. In exams, remember that “at least” corresponds to \(\ge\), while “at most” corresponds to \(\le\).

Substituting x = 2 into option C gives 2 ≤ 2, which is true because ≤ includes equality. The other options are false for x = 2.

The sign (\le) includes the boundary and smaller numbers. On the number line, smaller numbers are to the left.

‘At least 6’ means that \(x\) can be equal to 6 or greater than 6. Therefore, \(x\ge 6\) is correct. Option B, \(x\le 6\), means less than or equal to 6, which expresses the opposite direction. In exams, remember \(\ge\) for ‘at least’ and \(\le\) for ‘at most’.

‘At most 20’ means that \(n\) can be 20 or any value less than 20, so \(n\le 20\) is correct. The inequality \(n<20\) excludes 20, making it incorrect even though it is the closest distractor. Exam tip: use \(\le\) for ‘at most’ and \(\ge\) for ‘at least’.

Multiplying both sides of a > b by the positive number k does not reverse the inequality, so ak > bk is correct. Option B would be possible only if k were negative. Adding or subtracting the same number preserves the direction, so options C and D are also incorrect. Exam tip: multiplication or division by a positive number keeps the inequality sign unchanged; multiplication or division by a negative number reverses it.

Dividing both sides by (2) gives (3<x), that is (x>3). When rewriting, check the meaning of the sign direction.

From (2x-1<7), we get (2x<8) and (x<4), so (x=4) does not satisfy the strict inequality. Check strict signs carefully while testing options.

The possible integers are (3,4,5,6), so there are (4) values. The sign (\le) includes the left endpoint but (<) excludes the right endpoint.