Introduction to inequalities

Correct answer: B. (-3p<-3q)

Multiplying by a negative number reverses the inequality sign. This is the most common medium-level mistake.

Correct answer: A. (x<4)

Subtract (5) first and then divide by (2) so (x<4). The sign reverses only when division is by a negative number.

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

After subtracting (7) we get (-4x\le 12) and dividing by (-4) reverses the sign to (x\ge -3). Pay special attention to negative coefficients.

Correct answer: A. closed dot at (5) and shading to the left

The correct answer is option A: a closed dot at 5 with shading to the left. The inequality \(x\le5\) means x can be 5 or any number smaller than 5. Because equality is allowed, the point 5 must be included, so we draw a filled or closed dot. Smaller numbers lie to the left on a horizontal number line, so shade leftward. Option A shows both required features. Option B uses an open dot, which would mean 5 is excluded, as in \(x<5\). Option C uses a closed dot correctly for inclusion but shades right, representing numbers greater than or equal to 5. Option D uses an open dot and shades right, representing \(x>5\). Thus only A matches the inequality. Memory cue: filled means included; left means less than. The symbol \(\le\) combines both ideas.

Correct answer: B. ([3,8))

The value (3) is included so use ([ \ ) and (8) is not included so use (\ )). Check equality at endpoints.

Correct answer: C. (x>9)

Adding (2) gives (\frac{x}{3}>3) and multiplying by (3) gives (x>9). Multiplication by a positive number keeps the sign unchanged.

Correct answer: B. (m-6\le n-6)

Subtracting the same number from both sides keeps the inequality type unchanged. The symbol (\le) still allows equality.

Correct answer: B. (4)

The solution is (2x\ge 8) so (x\ge 4) and the least integer is (4). First find the real solution and then choose the integer.

Correct answer: A. (x<3)

From (7-2x>1) we get (-2x>-6) and dividing by (-2) gives (x<3). The sign reverses during negative division.

Correct answer: B. ([-1,\infty))

The direct answer is option B, \\([-1,\\infty)\\). The inequality \\(x\\ge-1\\) means x is -1 or any number greater than -1. In interval notation, a square bracket at -1 shows that -1 is included, and infinity always uses a parenthesis because infinity is not an actual number that can be reached. Therefore the interval is \\([-1,\\infty)\\). Option B is correct. Option A, \\(( -\\infty,-1]\\), describes numbers less than or equal to -1, so it reverses the direction. Option C, \\((-1,\\infty)\\), excludes -1, but the symbol \\(\\ge\\) includes it. Option D, \\((-\\infty,-1)\\), describes numbers strictly less than -1 and excludes the endpoint. Check the endpoint first: equality means a square bracket; “greater than” means the interval extends right.

Correct answer: A. (x\le 4)

Since (5) is positive, dividing by (5) does not change the sign and gives (x\le 4). Keep the equality part of the sign.

Correct answer: B. ac > bc

The governing rule is that multiplication or division by a positive number preserves the direction of an inequality. From a > b and c > 0, multiplying every part by c gives ac > bc, so option B is always true. The sign would reverse only if c were negative; that condition is explicitly excluded. Option A has the wrong direction. Dividing by the positive number c also preserves the direction, so option C is false because it uses < rather than >. Adding c to both sides would give a + c > b + c, not option D. Thus the positive-multiplier property identifies B without needing particular numerical values for a, b, or c.

Correct answer: A. (x\ge 4)

Subtracting (x) and (2) gives (2x\ge 8) so (x\ge 4). Dividing by positive (2) does not change the sign.

Correct answer: A. (-3\le x<4)

Subtracting (1) from all parts gives (-6\le 2x<8), and dividing by (2) gives (-3\le x<4). Change both bounds together in a compound inequality.

Correct answer: A. open dot at (2) and shading to the left

In (x<2), the value (2) is not included and numbers smaller than (2) lie to the left. Hence an open dot with left shading is correct.

Correct answer: A. (x+8<20)

The statement says add and less than, so it becomes (x+8<20). While translating words to symbols, identify the operation carefully.

Correct answer: B. (3y\le 15)

The correct answer is option B: \(3y\le15\). The phrase “three times a number y” means multiply y by 3, giving \(3y\). The phrase “not more than 15” means the quantity may be less than 15 or exactly 15; in symbols, this is \(\le15\). Combining the phrases gives \(3y\le15\). Option A, \(3y<15\), would mean strictly less and would wrongly exclude equality, such as \(3y=15\). Option B is correct because “not more than” includes equality. Option C, \(3y>15\), means more than 15, the opposite meaning. Option D, \(3y\ge15\), means at least 15 and is also opposite to the intended restriction. Memory cue: “not more than” and “at most” mean \(\le\); “not less than” means \(\ge\).

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

Subtracting (6) gives (-3x\le 6), and dividing by (-3) gives (x\ge -2). Reverse the sign in negative division.

Correct answer: C. (x=3)

The inequality gives (2x<8), so (x<4). Among the given values (0,2,3,-1), all satisfy it, so no listed value is not a solution.

Correct answer: A. (x\le 8)

Multiplying by (3) gives (2x-1\le 15), then (2x\le 16), so (x\le 8). Multiplication by a positive denominator does not change the sign.

Correct answer: D. (x) is greater than or equal to (12)

The symbol (\ge) means greater than or equal to. Accurate reading of symbols is essential in word-based questions.

Correct answer: A. \(x+4<y+4\)

Adding the same number, 4, to both sides of an inequality does not change its direction. Therefore, \(x<y\) implies \(x+4<y+4\). Option B is incorrect because adding a number does not reverse the order. Exam tip: adding or subtracting the same quantity on both sides preserves the inequality sign.

Correct answer: A. always true

The square of a real number is never negative, so (x^2\ge 0) is always true. This shows a basic idea of inequalities.

Correct answer: C. (x\le 5)

The symbol (\le) includes equality, so (x=5) is allowed. Strict signs (<) and (>) do not include the boundary.

Correct answer: B. (x\le -5)

Dividing by (-2) reverses the inequality, so (x\le -5). In questions with a negative coefficient, changing the sign is essential.