Inequality strategies
58 flashcards covering Inequality strategies for the GMAT Quantitative section.
Inequality strategies involve comparing two expressions using symbols like greater than (>), less than (<), or equal to (≥ or ≤). These problems require finding the range of values that satisfy the inequality, often through algebraic manipulation, such as adding or subtracting terms, multiplying or dividing by positive or negative numbers, or graphing on a number line. Mastering these strategies is essential for solving real-world scenarios in math, as they help determine feasible solutions under constraints.
On the GMAT Quantitative section, inequalities appear in problem-solving and data sufficiency questions, testing your ability to manipulate expressions and identify solution sets. Common traps include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, or overlooking absolute values that create multiple cases. Focus on practicing algebraic techniques, testing boundary values, and interpreting inequalities in word problems to avoid errors and improve accuracy.
Remember to always double-check the inequality direction after any multiplication or division by a negative.
Terms (58)
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Inequality
An inequality is a mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, indicating that one is less than, greater than, equal to or less than, or equal to or greater than the other.
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Linear inequality
A linear inequality involves a linear expression and compares it to a value, such as 2x + 3 > 5, and solving it means finding the values of x that make the inequality true.
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Compound inequality
A compound inequality combines two inequalities with 'and' or 'or', like -2 < x < 3, which means x is greater than -2 and less than 3 simultaneously.
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Adding inequalities
When adding two inequalities, you can add their corresponding sides as long as the inequalities point in the same direction, preserving the inequality sign.
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Subtracting inequalities
Subtracting inequalities works like adding them, but you subtract corresponding sides, ensuring the direction of the inequality remains the same.
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Multiplying inequality by positive number
If you multiply both sides of an inequality by a positive number, the inequality sign stays the same, such as multiplying 2x > 4 by 3 to get 6x > 12.
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Multiplying inequality by negative number
Multiplying both sides of an inequality by a negative number requires flipping the inequality sign, for example, multiplying -2x < 4 by -1 gives 2x > -4.
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Dividing inequality by positive number
Dividing both sides of an inequality by a positive number keeps the inequality sign unchanged, like dividing x/2 > 3 by 2 to get x > 6.
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Dividing inequality by negative number
Dividing both sides of an inequality by a negative number flips the inequality sign, such as dividing -x > 4 by -1 to get x < -4.
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Transitive property of inequalities
The transitive property states that if a > b and b > c, then a > c, which can be used to compare values indirectly in inequality problems.
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Absolute value inequality
An absolute value inequality involves expressions like |x - 2| > 3, which means the distance between x and 2 is greater than 3, so x < -1 or x > 5.
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Solving |x| > a
For |x| > a where a > 0, the solution is x < -a or x > a, representing values where x is farther from zero than a.
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Solving |x| < a
For |x| < a where a > 0, the solution is -a < x < a, meaning x is closer to zero than a.
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Quadratic inequality
A quadratic inequality compares a quadratic expression to zero, like x² - 4x + 3 > 0, and solving it involves finding the roots and testing intervals.
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Sign chart for inequalities
A sign chart is a tool to determine where a quadratic or polynomial expression is positive or negative by plotting roots on a number line and testing intervals.
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AM-GM inequality
The arithmetic mean-geometric mean inequality states that for non-negative numbers, the arithmetic mean is at least the geometric mean, such as (a + b)/2 ≥ √(a b) for a, b ≥ 0.
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Equality condition in AM-GM
In the AM-GM inequality, equality holds when all the numbers are equal, meaning for (a + b)/2 = √(a b), a must equal b.
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System of inequalities
A system of inequalities is a set of two or more inequalities that must all be true simultaneously, and solving it involves finding the overlapping solution region.
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Graphing inequalities
Graphing an inequality on a coordinate plane involves shading the region that satisfies the inequality, using a dashed line for strict inequalities and a solid line for inclusive ones.
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Feasible region
In a system of inequalities, the feasible region is the area on the graph where all inequalities overlap, representing all possible solutions.
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Testing values in inequalities
Testing values means plugging in numbers from different intervals into an inequality to see if they satisfy it, helping to identify the correct solution set.
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Common trap: dividing by variable
A common error is dividing both sides of an inequality by a variable without knowing its sign, which can lead to flipping the inequality incorrectly if the variable is negative.
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Common trap: squaring both sides
Squaring both sides of an inequality can change the direction if both sides are negative, so it's only safe if both sides are positive.
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Inequality with fractions
When solving inequalities with fractions, multiply both sides by the reciprocal of the denominator, flipping the sign if the denominator is negative.
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Inequalities in word problems
In word problems, inequalities translate phrases like 'at least' to ≥ or 'more than' to >, and solving them involves setting up and solving the inequality to find the range of values.
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Maximizing or minimizing with inequalities
To maximize or minimize a quantity subject to inequalities, identify the boundary points and evaluate the objective function at those points.
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Discriminant in quadratic inequalities
The discriminant (b² - 4ac) helps determine the nature of roots in a quadratic inequality; if positive, two real roots; if zero, one root; if negative, no real roots.
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Strict vs. non-strict inequalities
Strict inequalities use < or > and do not include equality, while non-strict ones use ≤ or ≥ and include the boundary values.
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Reciprocal inequality
For positive numbers, if a > b > 0, then 1/a < 1/b, meaning the inequality sign flips when taking reciprocals of positive values.
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Inequalities with roots
When dealing with square roots in inequalities, ensure the expression inside is non-negative and consider the domain, as square roots are defined only for non-negative numbers.
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Example: Solve 2x + 3 > 7
To solve 2x + 3 > 7, subtract 3 from both sides to get 2x > 4, then divide by 2 to get x > 2.
For x = 3, 2(3) + 3 = 9 > 7 is true; for x = 1, 2(1) + 3 = 5 > 7 is false.
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Example: Solve |x - 1| ≤ 2
To solve |x - 1| ≤ 2, it means -2 ≤ x - 1 ≤ 2, so add 1 to all parts: -1 ≤ x ≤ 3.
x = 0 is in the solution since |0 - 1| = 1 ≤ 2.
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Example: Solve x² - 4 < 0
To solve x² - 4 < 0, factor to (x - 2)(x + 2) < 0, find roots at x=2 and x=-2, and test intervals to get -2 < x < 2.
For x=0, 0² - 4 = -4 < 0 is true.
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Strategy: Isolate variable
In inequality problems, always isolate the variable on one side by performing operations that maintain the inequality's direction.
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Strategy: Consider domain restrictions
When solving inequalities with variables in denominators or under roots, consider domain restrictions to exclude values that make expressions undefined.
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Strategy: Use number line
Drawing a number line helps visualize inequality solutions by marking critical points and shading the appropriate regions.
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Strategy: Check endpoints
For inequalities with ≤ or ≥, check if the endpoints are included in the solution set by testing them in the original inequality.
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Inequality chains
An inequality chain links multiple comparisons, like a > b > c, which must all hold true simultaneously.
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Common trap: Assuming equality
A frequent mistake is assuming an inequality implies equality, but inequalities show a range, not a single value.
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Weighted AM-GM
The weighted AM-GM inequality extends the basic version for numbers with weights, stating that the weighted arithmetic mean is at least the weighted geometric mean.
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Cauchy-Schwarz inequality
Cauchy-Schwarz inequality for two sequences states that the square of the sum of products is less than or equal to the product of the sums of squares, useful in optimization.
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Jensen's inequality
Jensen's inequality applies to convex functions, stating that for a convex function f, f of the average is less than or equal to the average of f.
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Inequalities in functions
Inequalities can compare function values, like f(x) > g(x), requiring analysis of the functions' behaviors and intersections.
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Monotonic functions and inequalities
For monotonic increasing functions, if a > b, then f(a) > f(b), which can simplify inequality proofs.
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Example: AM-GM with two numbers
For numbers 4 and 9, AM-GM gives (4 + 9)/2 ≥ √(49), so 6.5 ≥ 6, with equality when 4=9.
The arithmetic mean is 6.5, geometric mean is 6.
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Example: System of two inequalities
Solve x + y > 5 and x - y < 3 by graphing or algebraically to find the region above one line and below the other.
For x=4, y=2: 4+2=6>5 and 4-2=2<3, so it's a solution.
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Strategy: Substitution in inequalities
Substitute expressions to simplify inequalities, ensuring the substitution doesn't alter the inequality's validity.
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Common trap: Overlooking extraneous solutions
In inequalities involving even roots, solutions might not satisfy the original due to domain restrictions, so always verify.
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Inequalities with exponents
For exponential inequalities like 2^x > 8, rewrite as x > 3 since 8=2^3, considering the base is greater than 1.
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Logarithmic inequalities
For logarithmic inequalities like log(x) > 1 with base greater than 1, solve by exponentiating, remembering the domain x > 0.
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Example: Solve 1/x > 2 for x
For 1/x > 2 and x > 0, multiply both sides by x to get 1 > 2x, then divide by 2: 0.5 > x, or x < 0.5.
x=0.4 satisfies since 1/0.4=2.5>2.
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Strategy: Factor and analyze
Factor expressions in inequalities to find critical points and determine sign changes across those points.
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Inequalities in sequences
Inequalities can compare terms in sequences, like in arithmetic or geometric sequences, to find patterns or bounds.
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Bounded regions in inequalities
Some systems of inequalities create bounded regions, like polygons, where maximum and minimum values occur at vertices.
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Example: Quadratic inequality with discriminant
For x² + 2x - 3 ≤ 0, discriminant is 4 + 12 = 16, roots are -3 and 1, so solution is -3 ≤ x ≤ 1.
x=0 satisfies since 0 + 0 - 3 = -3 ≤ 0.
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Strategy: Use symmetry
In symmetric inequalities, like those involving absolute values, exploit symmetry to simplify solving.
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Common trap: Misinterpreting 'and' vs. 'or'
In compound inequalities, 'and' means intersection of solutions, while 'or' means union, and confusing them leads to incorrect ranges.
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Inequalities with absolute values and quadratics
Combine absolute values and quadratics by solving the inner expressions first and then applying inequality rules.