Practice Questions
Concept Explanation
An inequality compares two expressions using <, >, ≤, or ≥. Solving finds the range of values that satisfy the relation.
Rules
- Add or subtract the same number on both sides — direction unchanged.
- Multiply or divide by a negative number — reverse the inequality sign.
- Flip when taking reciprocals of negative quantities.
Example
−3x < 9 ⇒ x > −3 (divide by −3, flip sign).
Common Trap
Multiplying or dividing by a negative without flipping the sign is the most common error.
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Q1 Easy
Find all values of x such that x + 3 > 7.
Explanationx > 4. -
Q2 Easy
Find all values of x such that 2x − 5 ≤ 9.
Explanation2x ≤ 14, so x ≤ 7. -
Q3 Medium
Find all values of x such that −4x > 12.
ExplanationDivide by −4 and flip: x < −3. -
Q4 Easy
Find all values of x such that 3(x − 2) ≥ 6.
Explanationx − 2 ≥ 2, so x ≥ 4. -
Q5 Easy
Which of the following values of x satisfies both x > 1 and x < 5?
Explanation3 lies strictly between 1 and 5. -
Q6 Medium
Find all values of x such that 5 − 2x ≥ 1.
Explanation−2x ≥ −4, so x ≤ 2. -
Q7 Medium
Find all values of x such that (x + 1)/2 < 4.
Explanationx + 1 < 8, so x < 7.
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Q8 Medium
If 2x + 3 > 11, what is the smallest possible integer value of x?
Explanation2x > 8, x > 4. Smallest integer is 5. -
Q9 Medium
Find all values of x such that x² ≤ 9.
Explanationx² ≤ 9 gives −3 ≤ x ≤ 3. -
Q10 Hard
Find all values of x such that |x − 2| < 3.
Explanation|x − 2| < 3 means −3 < x − 2 < 3, so −1 < x < 5. -
Q11 Hard
For what values of x is 1/x > 0?
ExplanationPositive reciprocal requires x > 0. -
Q12 Medium
Find all values of x such that 4x + 1 > 2x + 9.
Explanation2x > 8, so x > 4. -
Q13 Hard
Find all values of x such that −3 ≤ 2x + 1 < 7.
ExplanationSubtract 1: −4 ≤ 2x < 6. Divide by 2: −2 ≤ x < 3. -
Q14 Easy
If x is an integer and x < 4, what is the greatest possible value of x?
ExplanationGreatest integer strictly less than 4 is 3.
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DS1 Easy
Can it be determined whether x is greater than 5?
Statement I2x > 12.
Statement IIx − 3 > 2.
ExplanationEach alone: 2x > 12 gives x > 6; x − 3 > 2 gives x > 5. -
DS2 Medium
What inequality is equivalent to −2x < 6 after it is fully simplified?
Statement Ix is an integer.
Statement IIDividing by −2 gives x > −3.
ExplanationStatement II alone gives x > −3.
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DS3 Easy
Can it be determined whether x is less than or equal to 4?
Statement I3x ≤ 12.
Statement IIx + 1 ≤ 5.
ExplanationEach alone gives x ≤ 4. -
DS4 Easy
Does the value x = 4 satisfy both x > 1 and x < 6?
Statement Ix = 4.
Statement II4 is between 1 and 6.
ExplanationEach alone confirms x = 4 satisfies both conditions.