CA Foundation · Quantitative Aptitude

Linear Inequalities: formula sheet

Full chapter guide

Key formulas

Strict inequalities: a < b or a > b; Equality is not allowed. The boundary value is excluded, shown by an open circle on the number line.
Slack inequalities: a ≤ b or a ≥ b; Equality is allowed. The boundary value is included, shown by a filled circle.
Linear inequality in one variable: ax + b < 0 (or >, ≤, ≥), where a ≠ 0; Highest power of x is 1.
Quadratic inequality in one variable: ax² + bx + c < 0 (or >, ≤, ≥), where a ≠ 0; Highest power of x is 2.
Linear inequality in two variables: ax + by ≤ c (or <, >, ≥); Its solution is a region of the plane, not a single point.
Double inequality: p < x ≤ q; Means x > p and x ≤ q together. Check each end's symbol separately.
Addition rule: If a < b, then a + c < b + c; True for any real number c, positive or negative. The sign does not change.
Subtraction rule: If a < b, then a - c < b - c; True for any real number c. The sign does not change.
Multiplication or division by a positive number: If a < b and c > 0, then ac < bc and a ÷ c < b ÷ c; The sign stays the same.
Multiplication or division by a negative number: If a < b and c < 0, then ac > bc and a ÷ c > b ÷ c; The sign reverses. < becomes >, ≤ becomes ≥, and the other way round.
Transposing a term: x + a < b ⇒ x < b - a; Moving a term across the sign changes its sign. The inequality sign stays the same.
Interval notation: x > a is (a, ∞); x ≥ a is [a, ∞); x < a is (-∞, a); x ≤ a is (-∞, a]; Infinity always takes a round bracket.
Boundary line: Replace the sign in ax + by (<, >, ≤, ≥) c with = to get ax + by = c; Draw this line first. It separates the two half-planes.
Line style: ≤ or ≥: solid line. < or >: dashed line; Solid means points on the line are solutions. Dashed means they are not.
Test point rule: Put (x₀, y₀) not on the line into the inequality. True: shade its side. False: shade the opposite side.; If c ≠ 0, (0, 0) is the easiest test point. If c = 0, the line passes through the origin, so do not use (0, 0).
Intercepts of the line: x-intercept = c ÷ a (put y = 0). y-intercept = c ÷ b (put x = 0).; Use for a ≠ 0, b ≠ 0 and c ≠ 0. Then two intercepts are enough to draw the line. If c = 0, both intercepts are the origin, so find another point, such as the one at x = 1.
Special lines: x = k is a vertical line. y = k is a horizontal line.; x > k shades to the right of x = k. y > k shades above y = k.
Boundary line: ax + by ≤ c → boundary line: ax + by = c; Replace the inequality sign with = to get the line you draw.
Line style: ≤ or ≥: solid line. < or >: dashed line; A solid line means points on the line are included. A dashed line means they are not.
Test-point rule: Put (0, 0) into the inequality. True: shade the side with the origin. False: shade the other side.; Works only if the line does not pass through the origin. If it does, test another point such as (1, 0) or (0, 1).
Feasible region: Feasible region = intersection of all the shaded half-planes; A point is in the region only if it satisfies every inequality.
Corner point: Solve the two boundary equations together; Check the result in all other inequalities. A real corner point must satisfy them all.
Axis conditions: x ≥ 0 means right of or on the y-axis. y ≥ 0 means above or on the x-axis.; Together they confine the region to the first quadrant.
Phrase to symbol: at most / not more than / maximum → ≤ ; at least / not less than / minimum → ≥; Strict signs < and > are used for 'less than' and 'more than'.
Total cost model: Total cost = Fixed cost + (Variable cost per unit × x); Use this when a cost limit is given with a fixed part.
Profit model: Profit = Revenue − Cost = (Selling price × x) − (Fixed cost + Variable cost × x); For a profit target, set Profit ≥ target.
Negative multiplier rule: If a > b, then −a < −b (multiplying or dividing by a negative reverses the sign); Most common sign error in solving.
Resource constraint: a₁x + a₂y ≤ available resource; a₁ and a₂ are resource use per unit of two products.
Average condition: (Sum of values) ÷ n ≥ required average, so Sum ≥ required average × n; Used in marks-type and sales-type problems.

Quick revision

Symbols: < less than, > greater than, ≤ at most, ≥ at least.
Adding or subtracting the same number on both sides keeps the sign.
Multiplying or dividing both sides by a positive number keeps the sign.
Multiplying or dividing both sides by a negative number reverses the sign.
Strict inequalities (< or >) exclude the end value; round bracket ( ) in interval form.
Slack inequalities (≤ or ≥) include the end value; square bracket [ ] in interval form.
Always use round brackets for infinity, never square.
Boundary line of ax + by < c or > c is dashed; for ≤ or ≥ it is solid.
Test (0, 0) to find the shaded side, unless the line passes through it.
For a system, the solution is the common region of all inequalities.
In word problems, add x ≥ 0 and y ≥ 0 when quantities cannot be negative.
Check your answer by substituting one value from the solution set.

Common mistakes

Treating x > 3 as if 3 is a solution. Fix: Remember: only ≤ and ≥ include the boundary. Substitute 3 and see if 3 > 3 is true. It is not.
Calling x² + 3x ≥ 0 a linear inequality because it looks simple. Fix: Always find the highest power of the variable. Power 2 means quadratic.
Not reversing the sign when dividing by a negative number, for example -3x < 12 ⇒ x < -4. Fix: Circle any negative coefficient before dividing. Here -3x < 12 gives x > -4.
Reversing the sign when adding or subtracting a negative number. Fix: Only multiplication and division by a negative number reverse the sign. Adding or subtracting never does.
Using a solid line for a strict inequality, or a dashed line for ≤ or ≥. Fix: Remember: the 'equal to' bar in ≤ and ≥ means the line is included, so draw it solid.
Using (0, 0) as the test point when the line passes through the origin. Fix: If c = 0, the origin is on the line. Use another point such as (1, 0) or (0, 1).
Shading the wrong side of a line Fix: Always test (0, 0) when it is not on the line. If it makes the inequality true, shade its side. Otherwise shade the opposite side.
Testing a point on the boundary line Fix: When the line passes through the origin, use a point such as (1, 0) or (0, 1) instead.
Choosing the wrong symbol for 'at least' or 'at most'. Fix: Remember that 'at least' means the smallest allowed value, so the quantity can be that or more: ≥. Underline the phrase before writing the sign.
Not reversing the sign when dividing by a negative number. Fix: Whenever the coefficient of x is negative, pause and flip the sign after dividing. Better, move terms so that x has a positive coefficient.

Exam tips

Questions on this topic are usually quick classification or substitution checks. Do them fast and save time for longer sums.
Always test the boundary value when the options include it. This is the most common trap.
Identify degree by the highest power, not by how complex the expression looks.
If two options differ only in an open or closed endpoint, the strict or slack symbol decides the answer.
Questions often hide a negative coefficient. Look for it first, because the options usually include both directions of the sign.
Check the brackets in interval options. Two options may have the same numbers but one includes the end value and one does not.
For questions asking for the greatest or least integer, solve the inequality first, then pick the integer. Check whether the boundary value is allowed.
Test one value from your answer in the original inequality. It takes about ten seconds and catches most sign errors.