Introduction to inequalities

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.

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.

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.

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.

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.

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.

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.

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