Introduction to inequalities

Correct answer: A. (x\le -7)

From (2x+3\le x-4), we get (x\le -7). In exams keep (x)-terms on one side and constants on the other side.

Correct answer: A. (x\ge 7)

Since (3) is positive, the sign does not change, and (x-1\ge6) gives (x\ge7). In exams first check the sign of the denominator.

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

From (2x\le8), (x\le4), and with (x\in\mathbb{N}) we get ({1,2,3,4}). In exams always check the domain.

Correct answer: A. (2<x<8)

Both conditions are required together, so (2<x<8). In exams and usually means intersection (\cap).

Correct answer: A. (x\ge2)

(-3x\le -6), and dividing by (-3) gives (x\ge2). In exams reverse the sign when dividing by a negative coefficient.

Correct answer: A. (p+r<q+r) for every real (r)

Adding the same real number does not change the direction of inequality. In exams the sign of the multiplier or divisor matters.

Correct answer: A. (mr<nr)

Multiplying by negative (r) changes (>) into (<). In exams watch for reversal in options containing (r<0).

Correct answer: A. (x\le -3) or (x\ge3)

In (|x|\ge3), the distance from (0) is at least (3). In exams a greater-than absolute value gives two outer parts.

Correct answer: A. always true

The square of a real number is never negative, so (x^2\ge0) is always true. In exams remember the basic property of squares.

Correct answer: A. (\varnothing)

For real (x), (x^2) is never negative. In exams write (\varnothing) for an impossible inequality.

Correct answer: A. (x<-3) or (x>2)

The product is positive when both factors have the same sign. In exams keep the zero points (-3) and (2) open.

Correct answer: A. (x<-1) or (x>2)

The fraction is positive when numerator and denominator have the same sign. In exams (x=-1) makes the denominator zero, so it is excluded.

Correct answer: A. (3)

At (x=3), the denominator becomes (0), so the fraction is undefined. In exams always exclude values that make the denominator zero.

Correct answer: A. (\mathbb{R})

After cancelling (x), we get (2>-5), which is always true. In exams such questions may have all real numbers as the answer.

Correct answer: A. \(\varnothing) / (\varnothing)

After cancelling (x), we get (-6\ge1), which is false. In exams a false constant statement means no solution.

Correct answer: A. (2<x\le6)

Adding (3) to all three parts gives (4<2x\le12), so (2<x\le6). In exams apply each step to all three parts.

Correct answer: A. (m\ge40)

Greater than or equal to is written as (m\ge40). In exams when the word equal appears, look for (\ge) or (\le).

Correct answer: A. (w\le500)

The direct answer is A: \(w\le500\). The phrase maximum load of 500 kg means the lift may carry 500 kg or any smaller non-negative load. “Maximum” includes the limiting value, so the correct symbol is \(\le\), read as less than or equal to. Option A correctly allows w=500 as well as smaller values. Option B, \(w<500\), wrongly excludes exactly 500 kg; it would describe a strict limit below 500. Option C, \(w\ge500\), says the load must be at least 500 kg, which is not the meaning of a maximum and would allow heavier loads. Option D, \(w>500\), says the load must exceed the safe limit and is clearly unsafe. Physically, w cannot be negative, but that extra fact is not needed among the options. Memory cue: maximum means “at most,” so use \(\le\); minimum means “at least,” so use \(\ge\).

Correct answer: A. (3x<21)

The direct answer is A: \(3x<21\). Translate each phrase carefully. “A number” is x. “Three times a number” means multiplying x by 3, giving 3x. “Is less than 21” gives the symbol <, so the complete inequality is \(3x<21\). It could also be solved as x<7 by dividing by positive 3, but the question asks for the correct statement and A is the direct translation. Option B, \(3x>21\), reverses “less than” and says greater than. Option C, \(x+3<21\), means 3 is added to x, not that x is multiplied by 3. Option D, \(x/3<21\), means one-third of x is less than 21, which is a different sentence. The order matters: “three times x” is 3x, not x+3 or x/3. Memory cue: times means multiplication; less than means the left expression is smaller.

Correct answer: A. (x=6)

The direct answer is A: \(x=6\). Solve first: \(2x-3>7\). Add 3 to both sides: \(2x>10\). Divide by positive 2: \(x>5\). Therefore any value greater than 5 works. Test the options directly: for x=6, \(2(6)-3=12-3=9>7\), so A satisfies the inequality. For x=5, the left side is \(10-3=7\), giving 7>7, which is false; the boundary is not included, so B fails. For x=4, the left side is 5, not greater than 7, so C fails. For x=3, the left side is 3, so D fails. The strict sign > is important: x must be strictly greater than 5, not equal to 5. Memory cue: after solving, substitute each offered value into the original inequality, not a changed form.

Correct answer: A. (x=4)

From (5-4x\ge-7), (x\le3), so (x=4) is not in the solution. In exams notice the phrase does not satisfy.

Correct answer: A. (x\le4)

From (x+3\ge2x-1), (4\ge x), that is (x\le4). In exams understand the direction when rewriting (4\ge x) as (x\le4).

Correct answer: A. (-2\le x<8)

Multiplying all parts by positive (2) gives (-2\le x<8). In exams positive multiplication does not change signs.

Correct answer: A. (-5\le x<3)

Dividing by (-2) reverses both signs and gives (3>x\ge-5), that is (-5\le x<3). In exams rewrite compound inequalities in increasing order.

Correct answer: A. (-14<x\le4)

Multiplying by positive (3) gives (-12<x+2\le6), so (-14<x\le4). In exams preserve open and closed signs separately.