Introduction to inequalities

From (7-3x\ge 1), we get (-3x\ge -6), and dividing by (-3) gives (x\le 2). The sign reverses when dividing by a negative number.

The direct answer is option A, \\(1<x\\le6\\). Interval notation uses a round parenthesis when an endpoint is excluded and a square bracket when it is included. Thus the left parenthesis in \\( (1,6] \\) excludes 1, giving \\(x>1\\), while the right bracket includes 6, giving \\(x\\le6\\). Combining these gives \\(1<x\\le6\\). Option A is correct. Option B, \\(1\\le x<6\\), includes the wrong endpoint and excludes 6. Option C, \\(1<x<6\\), correctly excludes 1 but wrongly excludes 6. Option D, \\(1\\le x\\le6\\), includes both endpoints, which does not match the notation. Test the endpoints directly: 1 is out, 6 is in. A bracket means included; a parenthesis means excluded.

At least (25) means (n\ge 25), and at most (40) means (n\le 40). Combining both gives (25\le n\le 40).

First subtract (3) from both sides to get (6<3x), then divide by (3) to get (x>2). When rewriting an inequality, always check the meaning of its direction.

The sign (>) means the value on the left is greater. In exams the open side of the sign faces the greater number.

The governing concept is the meaning of the symbol ≤. It means “less than or equal to,” so equality at the boundary is allowed. Substitute y = -3 into the statement: -3 ≤ -3. The two sides are equal, and an equality is included by ≤; therefore the statement is true and option A is correct. The statement would be false if the sign were <, because -3 is not strictly less than -3. Option B incorrectly treats ≤ as a strict inequality. Options C and D are unsuitable because the value of y is already given, so the truth value can be determined directly and there is no issue of an empty solution set.

In (5x-2\ge 13), the variable has power (1), so it is a linear inequality. Forms with (x^2) or (xy) are not linear.

The number (4) is less than (5), so it does not satisfy (x\ge 5). Check options by substituting them directly.

“Less than” is represented by <, so “t is less than 11” is written as \(t<11\). Option A reverses the inequality and means “t is greater than 11,” while C and D mean “less than or equal to” and “equal to,” respectively. Exam tip: For “less than,” the variable must have the smaller value.

“Not less than 9” means “9 or more,” so the correct inequality is \(z\ge 9\). It includes equality, meaning \(z=9\) is also allowed. Option B is incorrect because \(z\le 9\) means less than or equal to 9. Exam tip: interpret “not less than” as “at least” and use \(\ge\).

An inequality compares two quantities and shows that one is greater or smaller than the other. When the same number is added to both sides, the relationship remains unchanged. Only the values on the two sides move by the same amount, so the greater side is still greater.

Starting with the given statement, add 5 to both sides: a+5>2+5. The right side becomes 7, giving a+5>7. Therefore, option A is correct. The direction does not reverse because a positive number was added, not multiplied by a negative number. Option B reverses the sign without reason, while C and D do not represent adding 5 to both sides.

An inequality compares two quantities and shows that one may be smaller than or equal to the other. A basic rule says that if the same number is added to or subtracted from both sides, the inequality sign keeps its direction. The sign changes direction only when both sides are multiplied or divided by a negative number.

Starting from b ≤ 12, subtract 4 from the left side and also from the right side. This gives b − 4 ≤ 12 − 4, and simplifying the right side gives b − 4 ≤ 8. Therefore, the stated result is correct. The alternatives with ≥ change the direction without justification, while b + 4 and 4b do not represent subtracting 4 from both sides.

Multiplying by a negative number reverses the inequality sign. While multiplying by (-2), changing the direction is necessary.

For (x=4), (2\cdot4+3=11), and (11\le 11) is true. A sign with equality accepts the boundary value.

The sign (\ge) includes the boundary, and greater numbers are to the right. So use a closed circle at (-1) and shade right.

Interval notation describes which real numbers belong to a set. In
(4,\infty), the round parenthesis at 4 means that 4 is not included. The symbol \(\infty\) indicates that the interval continues without an upper endpoint, so the set contains all numbers to the right of 4 on the number line. Therefore, every allowed value is greater than 4, not equal to 4.

To express this in inequality form, compare x with the left endpoint: \(x>4\). For example, 4.1, 5, and 100 belong to the interval, while 4 does not. A closed endpoint would use a square bracket and would allow equality, giving \(x\ge4\), but that is not the notation shown here. Thus option B correctly translates the interval.

The number (6) is greater than (5), so it is not included. The sign (\le) includes boundaries but not outside values.

In (r\le 0), (0) and smaller numbers are included. With an equality sign, the boundary value is also a solution.

‘At least \(33\)’ means \(33\) or more, so the correct inequality is \(m\ge 33\). Option A excludes exactly \(33\), while options C and D represent marks less than or equal to \(33\). Exam tip: associate ‘at least’ with \(\ge\) and ‘at most’ with \(\le\).

“Not more than 20 kilograms” means that the weight can be less than or equal to 20 kilograms. Therefore, the inequality is \(w\le 20\). Option A is incorrect because it excludes a weight of exactly 20 kilograms, whereas “not more than” includes equality. Exam tip: Use \(\le\) for phrases such as “at most” and “not more than.”

‘Less than’ means strictly smaller than 250, so the correct inequality is \(p<250\). Option B, \(p\le250\), also includes a price equal to 250, which is not stated in the question. In exams, remember that ‘less than’ uses \(<\), while ‘less than or equal to’ uses \(\le\).

“Not less than 16 years” means “at least 16 years,” so the age may be equal to 16 or greater: \(a\ge 16\). Option D is incorrect because \(a>16\) excludes the age of exactly 16 years. In an exam, translate “not less than” or “at least” as \(\ge\).

Substituting x = 7 into option C gives 7 > 4, which is true. Options A and B are false for x = 7, and option D is an equation, not an inequality (and 7 = 5 is false).

The direct answer is option A: \\(x\\in\\{1,2,3,4,5\\}\\). A natural number is taken here as a positive counting number: 1, 2, 3, and so on. The condition \\(x\\le5\\) allows every natural number up to and including 5. Therefore the complete set is \\(\\{1,2,3,4,5\\}\\). Option A is correct. Option B includes 0; under the convention used in this question, 0 is not a natural number. Option C begins at 2 and includes 6, so it omits 1 and violates \\(x\\le5\\). Option D omits 5, even though equality is allowed. The symbol \\(\\le\\) includes the endpoint. Always check the textbook convention for natural numbers, but here the supplied options and explanation clearly use the convention starting at 1.

“At most \(30\)” means that the value can be \(30\) or any number less than \(30\). Therefore, \(n\le 30\) is correct. Option B, \(n<30\), excludes the value \(n=30\). Exam tip: use \(\le\) for “at most” and \(\ge\) for “at least.”