12 multiple-choice questions, progressively harder.
Multiply both sides of the true statement −1<4-1 < 4−1<4 by −5-5−5. The correct result is:
Solution
Correct answer: A
Each side scales by −5-5−5: the left −1-1−1 becomes 555 and the right 444 becomes −20-20−20. The multiplier is negative, so reverse the direction.
−1<4 → ×(−5) 5>−20-1 < 4 \;\xrightarrow{\;\times(-5)\;}\; 5 > -20−1<4×(−5)5>−20
The result 5>−205 > -205>−20 is true.
Take one step to isolate xxx: −3x≥−12-3x \ge -12−3x≥−12.
Correct answer: C
Divide both sides by −3-3−3. The divisor is negative, so reverse ≥\ge≥ to ≤\le≤, and −12÷(−3)=4-12 \div (-3) = 4−12÷(−3)=4.
−3x−3≤−12−3 ⇒ x≤4\frac{-3x}{-3} \le \frac{-12}{-3} \;\Rightarrow\; x \le 4−3−3x≤−3−12⇒x≤4
A negative divided by a negative is positive.
The compound inequality −4≤x<0-4 \le x < 0−4≤x<0 in words is:
Correct answer: D
The left end ≤\le≤ is inclusive ('at least −4-4−4'), and the right end <<< is strict ('less than 000').
−4≤x<0 ⇒ at least −4 and less than 0-4 \le x < 0 \;\Rightarrow\; \text{at least } -4 \text{ and less than } 0−4≤x<0⇒at least −4 and less than 0
The boundary −4-4−4 is in, and 000 is out.
Take one step to isolate xxx: x+8≤3x + 8 \le 3x+8≤3.
Subtract 888 from both sides. Subtraction does not change the direction, even though the result is negative.
x+8−8≤3−8 ⇒ x≤−5x + 8 - 8 \le 3 - 8 \;\Rightarrow\; x \le -5x+8−8≤3−8⇒x≤−5
A negative answer does not trigger a flip.
Which interval is written incorrectly?
Infinity is not a number, so it can never take a bracket.
[3,∞] is wrong; it should be [3,∞)[3, \infty] \;\text{ is wrong; it should be } [3, \infty)[3,∞] is wrong; it should be [3,∞)
The other three use their fences correctly.
x≤7x \le 7x≤7 is the same statement as:
Swap the sides and reverse the symbol to read the inequality from the other end.
x≤7⟺7≥xx \le 7 \quad\Longleftrightarrow\quad 7 \ge xx≤7⟺7≥x
Both say 'xxx is at most 777.'
Which value satisfies 2<x≤62 < x \le 62<x≤6?
Correct answer: B
A solution must be greater than 222 and at most 666.
2<5≤6 ✓2 < 5 \le 6 \;\checkmark2<5≤6✓
The value 222 is excluded by the strict left end, 777 is above 666, and 111 is below 222, so only 555 works.
Reading −8<−3-8 < -3−8<−3 aloud gives:
The symbol <<< means 'less than', and −8-8−8 sits farther left than −3-3−3.
−8<−3 ⇒ negative eight is less than negative three-8 < -3 \;\Rightarrow\; \text{negative eight is less than negative three}−8<−3⇒negative eight is less than negative three
Among negatives, the one farther from zero is smaller.
Take one step to isolate xxx: x−3>1\dfrac{x}{-3} > 1−3x>1.
Multiply both sides by −3-3−3. The multiplier is negative, so reverse >>> to <<<, and 1⋅(−3)=−31 \cdot (-3) = -31⋅(−3)=−3.
−3⋅x−3<−3⋅1 ⇒ x<−3-3 \cdot \frac{x}{-3} < -3 \cdot 1 \;\Rightarrow\; x < -3−3⋅−3x<−3⋅1⇒x<−3
The flip comes from multiplying by a negative.
Is x=4x = 4x=4 a solution of −2x+3≥−6-2x + 3 \ge -6−2x+3≥−6? Substitute to check.
Substitute x=4x = 4x=4 into the left side and compare with the right.
−2(4)+3=−5,−5≥−6 ✓-2(4) + 3 = -5, \qquad -5 \ge -6 \;\checkmark−2(4)+3=−5,−5≥−6✓
Since −5≥−6-5 \ge -6−5≥−6 is true (it sits to the right of −6-6−6), x=4x = 4x=4 is a solution.
Which statement about ∞\infty∞ in interval notation is true?
Infinity is a direction, not a reachable number, so an interval is always open at an infinite end.
(2,∞)and(−∞,5](2, \infty) \quad\text{and}\quad (-\infty, 5](2,∞)and(−∞,5]
Every infinite end uses a parenthesis, never a bracket.
A number nnn is at least −2-2−2 and at most 444. In interval notation this is:
'At least −2-2−2' includes −2-2−2, and 'at most 444' includes 444, so both endpoints are inclusive.
−2≤n≤4⟺[−2,4]-2 \le n \le 4 \quad\Longleftrightarrow\quad [-2, 4]−2≤n≤4⟺[−2,4]
Both fences are brackets.
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