Introduction to inequalities

Correct answer: B. (-7 ≤ y)

The governing concept is the symmetry of an inequality under interchange of its two sides. The statement y ≥ -7 means that y is greater than or equal to -7. Reading the same comparison from right to left gives -7 ≤ y, which has exactly the same set of solutions. For example, y = -7 satisfies both statements, and y = 0 also satisfies both, while y = -8 satisfies neither. Option B is therefore correct. Option A reverses the comparison and represents y ≤ -7; options C and D also describe values at or below -7, not values at or above it. No sign reversal is needed merely because the two sides are written in reverse order; reversal occurs only when multiplying or dividing by a negative number.

Correct answer: B. a is less than or equal to b

The symbol ≤ means “less than or equal to.” It combines two possibilities: a<b and a=b. Thus a may be smaller than b, or it may have exactly the same value as b; equality is included because of the bar under the less-than sign. For example, if a=3 and b=5, the inequality is true because 3<5. If a=5 and b=5, it is also true because equality is allowed. Option B states both possibilities accurately. Option A incorrectly excludes equality, option C reverses the comparison, and option D contradicts the meaning of the inclusive symbol. On a number line, the corresponding endpoint would be included, usually shown with a closed point.

Correct answer: A. It is true for all x ∈ ℝ

The governing fact is that the square of every real number is non-negative: x² ≥ 0. Adding 4 to this inequality gives x² + 4 ≥ 4, so the statement holds for every real value of x. Equality occurs when x = 0, since 0² + 4 = 4; for every nonzero real x, x² is positive and the left side is greater than 4. Thus the result is universal, not restricted to positive or negative numbers. Option B excludes x = 0 and all negative values, while option C excludes zero and all positive values. Option D contradicts the basic non-negativity property of a square. Therefore option A is the only valid answer.

Correct answer: A. \(\{1,2,3,4,5,6\}\)

The question specifies that natural numbers start from 1. Therefore, the natural numbers satisfying \(x\le 6\) are \(1,2,3,4,5,6\). Since the inequality sign is \(\le\), the boundary value 6 is included in the solution set. Option B incorrectly includes 0, which is excluded under the convention stated here. In an exam, remember that \(\le\) and \(\ge\) include the boundary value.

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

Substituting \(x=5\) in option A gives \(x-1=4\), so \(4>5\) is false. The other options give \(6\ge6\), \(10\le10\), and \(5<8\), all of which are true. Therefore, A is correct. Exam tip: Substitute the given value first, then check whether the inequality sign remains valid.

Correct answer: A. \(x=3\) is a solution

Substituting \(x=3\) makes the left-hand side \(3(3)+4=13\). Thus, the inequality becomes \(13\le 13\), which is true because \(\le\) includes equality. Therefore, \(x=3\) is a solution. Option B is incorrect because it ignores the equality case. Exam tip: Always substitute a proposed value into the original inequality to verify whether it is a solution.

Correct answer: A. -4

For \(x\le -3\), any value less than or equal to \(-3\) is a solution. Since \(-4\le -3\) is true, option A is correct. The closest distractor, \(-2\), is greater than \(-3\), so it does not satisfy the inequality. In exams, use a number line to compare negative numbers accurately.

Correct answer: B. (2\le x<9)

The bracket ([2) shows (2) is included and (9)) shows (9) is not included. Hence (2\le x<9) is correct.