ORIGINAL WORKED EXAMPLE
Solve with a negative coefficient
Solve 7−3x≥19.
- Subtract 7: −3x≥12.
- Divide by −3 and reverse the symbol: x≤−4.
- At −4 equality holds; at −5 the original left side is 22≥19.
Answer x≤−4
GRE · Algebra · TOPIC LESSON
Inequalities, intervals and comparison
Treat inequalities as sets of allowed values and preserve their direction with care.
BY THE END, YOU CAN
- Solve and combine linear inequalities.
- Track sign reversal and endpoint inclusion.
- Use bounds and counterexamples for quantitative comparison.
BEFORE YOU START
- Linear equations
- Signed multiplication and division
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
Original GRE-style worked examples and practice
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
An inequality describes a region of possibilities
x<3 includes every real number below 3, not just integers. x≤3 also includes the boundary. A solution set may be an interval, a union of intervals, all real numbers or the empty set. State whether the variable is integer, positive, or otherwise restricted before interpreting the set.
A statement that is true for some allowed values is not necessarily true for all. This distinction is central to quantitative comparison and questions asking what must be true.
02 / UNDERSTAND & APPLY
Negative multiplication reverses the ordering
Adding or subtracting the same quantity preserves an inequality. Multiplying or dividing by a positive quantity preserves direction. Multiplying or dividing by a negative quantity reverses it: 2<5 becomes −2>−5. Multiplication by zero erases information and is not reversible.
If a multiplier contains a variable with unknown sign, do not multiply without splitting into sign cases. An inequality such as 1/x>2 cannot safely be turned into 1>2x unless x>0 is known.
03 / UNDERSTAND & APPLY
And means intersection; or means union
A compound inequality a<x≤b is the intersection of x>a and x≤b. Perform the same addition or subtraction on all three parts. If dividing the whole chain by a negative number, reverse both comparisons and rewrite the endpoints in increasing order.
“Or” combines sets: x<−2 or x≥4 gives two rays. A number needs to satisfy either condition. “And” requires both; disjoint conditions joined by and have no solutions.
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ORIGINAL WORKED EXAMPLE
A compound interval
Solve −5<2x+1≤9.
- Subtract 1 from all three parts: −6<2x≤8.
- Divide by positive 2: −3<x≤4.
- The left endpoint is excluded and the right included.
Answer (−3,4]
04 / UNDERSTAND & APPLY
Combine bounds only in valid directions
If a<b and c<d, then a+c<b+d. For products, sign information is essential. With 0<a<3 and 0<b<5, 0<ab<15. If a or b can be negative, the largest product may use two negative endpoints instead.
Subtracting inequalities is not the same as subtracting equations. To bound a−b, combine the smallest possible a with the largest possible b for a lower bound, and the largest a with the smallest b for an upper bound.
ORIGINAL WORKED EXAMPLE
Bound a difference
If 2<a<7 and −1<b<3, find the tight open interval for a−b.
- The smallest limiting difference uses a near 2 and b near 3, giving lower bound −1.
- The largest limiting difference uses a near 7 and b near −1, giving upper bound 8.
- Neither endpoint is attainable, so −1<a−b<8.
Answer (−1,8)
05 / UNDERSTAND & APPLY
Check signs before taking reciprocals or squares
For 0<a<b, reciprocals reverse order: 1/a>1/b. The same reversal holds if a<b<0, but an interval crossing zero cannot be inverted as one continuous interval because zero is excluded and the signs split.
Squaring preserves order for nonnegative quantities. It reverses magnitude comparisons for two negative quantities in the sense that the more negative number has the larger square. With mixed signs there is no single order rule without comparing magnitudes.
ORIGINAL WORKED EXAMPLE
A reciprocal inequality with an unknown sign
Solve 1/x>2 over the reals.
- Exclude x=0. If x<0, 1/x is negative and cannot exceed 2.
- If x>0, multiply by x without reversing: 1>2x.
- Combine with the sign condition: 0<x<1/2.
Answer (0,1/2)
06 / UNDERSTAND & APPLY
Count permitted integers only after solving
An interval solution over the reals can contain finitely many integers. List endpoints carefully. For −2<x≤3, the integers are −1,0,1,2,3, a total of five. Zero and negative integers are valid unless excluded by the wording.
For strict inequalities with noninteger endpoints, identify the first and last allowed integer. The count of consecutive integers from m through n inclusive is n−m+1. The plus one includes both ends.
ORIGINAL WORKED EXAMPLE
Count integer solutions
How many integers satisfy 3<2n−1≤13?
- Add 1 and divide by 2: 2<n≤7.
- The allowed integers are 3,4,5,6,7.
- The count is 7−3+1=5.
Answer 5
07 / UNDERSTAND & APPLY
A proof for all cases differs from a sample check
To choose A, B or C in QC, establish that relationship for every permitted input. To choose D, demonstrate two different relationships with allowed inputs. Equality versus strict inequality already counts as two different relationships; you do not need both A>B and A<B.
Useful test points include zero, negative numbers, fractions between 0 and 1, boundaries when included, and numbers on opposite sides of a critical point. Never use a convenient value that violates the prompt.
08 / UNDERSTAND & APPLY
Choose a method and answer the requested question
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
09 / UNDERSTAND & APPLY
Original practice: intervals and comparisons
Use the defined QC labels. Where an interval is requested, keep endpoints precise.
ORIGINAL PRACTICE · 1
Single answer: −4x<12 is equivalent to A x<−3; B x>−3; C x<3; D x>3; E x=−3.
Reveal answer & reasoning
B
Dividing by negative −4 reverses the symbol. Equality is excluded because the original inequality is strict.
ORIGINAL PRACTICE · 2
Numeric entry: How many integers satisfy −2≤x<4?
Reveal answer & reasoning
6
The integers are −2,−1,0,1,2,3. Including 4 or excluding −2 mishandles the boundary symbols.
ORIGINAL PRACTICE · 3
Select all that apply: If x>5, which must hold? A 2x>10; B −x<−5; C x²>25; D 1/x>1/5; E x−8>0.
Reveal answer & reasoning
A, B and C
A follows by positive multiplication, B by negative multiplication with reversal, and C by squaring positive values. Reciprocals give 1/x<1/5, not D. E fails for x=6.
ORIGINAL PRACTICE · 4
QC: 0<x<1. Quantity A: x². Quantity B: x³.
Reveal answer & reasoning
A
Multiplying x²>0 by x<1 decreases it, so x³<x². Integer-only trials would miss the given interval.
ORIGINAL PRACTICE · 5
QC: x≥2. Quantity A: x². Quantity B: 4.
Reveal answer & reasoning
D
At x=2 the quantities are equal; at x=3 A is greater. A≥B does not justify choice A, which requires strict greater-than for every permitted value.
ORIGINAL PRACTICE · 6
Single answer: x<1 and x≥4 has A all real solutions; B x<1; C x≥4; D no solutions; E 1≤x≤4.
Reveal answer & reasoning
D
No real number satisfies both disjoint conditions. Replacing “and” with “or” would change the problem.
ORIGINAL PRACTICE · 7
Numeric entry: What is the greatest integer satisfying 5x+2<28?
Reveal answer & reasoning
5
5x<26 gives x<5.2. The greatest integer is 5, which gives 27<28; 6 fails.
ORIGINAL PRACTICE · 8
QC: −4<a<−2. Quantity A: a². Quantity B: 4.
Reveal answer & reasoning
A
|a|>2, so a²>4. Although a itself is below −2, squaring compares magnitudes, not signed order.
KEEP THE BIG IDEAS
Quick revision
- Reverse the inequality when dividing or multiplying by a negative value.
- And intersects sets; or unites them.
- Unknown signs require cases before multiplying or reciprocating.
- QC choice D needs different allowed relationships, including equality versus inequality.
READ WITH CONTEXT
References & scope
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
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