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 EasyFind all values of x such that x + 3 > 7.Explanationx > 4.
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Q2 EasyFind all values of x such that 2x − 5 ≤ 9.Explanation2x ≤ 14, so x ≤ 7.
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Q3 EasyFind all values of x such that 3(x − 2) ≥ 6.Explanationx − 2 ≥ 2, so x ≥ 4.
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Q4 EasyWhich of the following values of x satisfies both x > 1 and x < 5?Explanation3 lies strictly between 1 and 5.
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Q5 EasyIf 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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Q6 MediumFind all values of x such that −4x > 12.ExplanationDivide by −4 and flip: x < −3.
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Q7 MediumFind all values of x such that 5 − 2x ≥ 1.Explanation−2x ≥ −4, so x ≤ 2.
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Q8 MediumFind all values of x such that (x + 1)/2 < 4.Explanationx + 1 < 8, so x < 7.
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Q9 MediumIf 2x + 3 > 11, what is the smallest possible integer value of x?Explanation2x > 8, x > 4. Smallest integer is 5.
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Q10 MediumFind all values of x such that x² ≤ 9.Explanationx² ≤ 9 gives −3 ≤ x ≤ 3.
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Q11 MediumFind all values of x such that 4x + 1 > 2x + 9.Explanation2x > 8, so x > 4.
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Q12 HardFind all values of x such that |x − 2| < 3.Explanation|x − 2| < 3 means −3 < x − 2 < 3, so −1 < x < 5.
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Q13 HardFor what values of x is 1/x > 0?ExplanationPositive reciprocal requires x > 0.
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Q14 HardFind all values of x such that −3 ≤ 2x + 1 < 7.ExplanationSubtract 1: −4 ≤ 2x < 6. Divide by 2: −2 ≤ x < 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 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.
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DS3 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. -
DS4 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.