Absolute value
51 flashcards covering Absolute value for the GMAT Quantitative section.
Absolute value represents the distance of a number from zero on the number line, without considering direction. For example, the absolute value of both 5 and -5 is 5, making it a non-negative value that helps in understanding magnitudes in math. It's a fundamental concept in algebra that simplifies working with equations and inequalities by focusing on the size of numbers rather than their signs.
On the GMAT Quantitative section, absolute value frequently appears in algebra problems, such as solving equations like |x + 3| = 7 or inequalities like |x - 2| < 4. These questions often test your ability to handle two cases for the expression inside the absolute value, which can lead to common traps like missing solutions or incorrectly assuming only positive values. Focus on practicing how to isolate the absolute value and verify answers to avoid errors in complex word problems. Remember to always consider both positive and negative scenarios when solving.
Terms (51)
- 01
Absolute value definition
The absolute value of a number is its distance from zero on the number line, resulting in a non-negative value.
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Absolute value notation
Absolute value is written as |x|, where x is the number, and it equals x if x is positive or zero, and -x if x is negative.
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Absolute value of zero
The absolute value of zero is zero, since it is already at zero on the number line.
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Absolute value properties
Absolute value has properties such as |a| ≥ 0 for any real number a, |a b| = |a| |b|, and |a + b| ≤ |a| + |b|, known as the triangle inequality.
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Triangle inequality
The triangle inequality states that for any real numbers a and b, the absolute value of their sum is less than or equal to the sum of their absolute values: |a + b| ≤ |a| + |b|.
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Solving |x| = c
To solve an equation like |x| = c where c > 0, the solutions are x = c or x = -c.
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Solving |x| > c
For an inequality like |x| > c where c > 0, the solution is x > c or x < -c.
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Solving |x| < c
For an inequality like |x| < c where c > 0, the solution is -c < x < c.
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Solving |x| ≥ c
For an inequality like |x| ≥ c where c > 0, the solution is x ≥ c or x ≤ -c.
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Solving |x| ≤ c
For an inequality like |x| ≤ c where c > 0, the solution is -c ≤ x ≤ c.
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Absolute value as distance
Absolute value represents the distance between two points on the number line, such as |a - b| being the distance between a and b.
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Graph of y = |x|
The graph of y = |x| is a V-shaped line with its vertex at the origin, consisting of y = x for x ≥ 0 and y = -x for x < 0.
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Nested absolute values
Nested absolute values, like ||x| - 2|, require evaluating the inner absolute value first before the outer one.
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Absolute value equations with no solution
An absolute value equation like |x| = -5 has no solution because absolute values are always non-negative.
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Absolute value inequalities with two cases
When solving inequalities like |ax + b| > c, consider two cases based on the expression inside the absolute value being positive or negative.
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Common trap: Ignoring cases in equations
A common error is solving absolute value equations without considering both positive and negative cases, leading to incomplete solutions.
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Absolute value in quadratic equations
Absolute value can appear in quadratic contexts, such as solving |x^2 - 1| = 2, which requires finding where the quadratic equals 2 or -2.
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Minimum value of |x - a|
The expression |x - a| reaches its minimum value of zero when x equals a, representing the closest point on the number line.
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Maximizing |x + y|
To maximize |x + y| for given constraints, consider the values of x and y that make x + y as large positive or large negative as possible.
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Absolute value of sums
The absolute value of a sum, |a + b|, is not necessarily equal to |a| + |b|, but it is always less than or equal to |a| + |b|.
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Absolute value of products
The absolute value of a product is the product of the absolute values: |a b| = |a| |b|.
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Absolute value inequalities with AND
In compound inequalities like |x - 3| < 2, it means -2 < x - 3 < 2, which combines both conditions simultaneously.
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Critical points in absolute value
Critical points in absolute value inequalities are the values where the expression inside the absolute value is zero, helping to determine intervals.
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Absolute value with fractions
For expressions like |1/x|, if x is not zero, it equals 1/|x|, since the absolute value of a reciprocal is the reciprocal of the absolute value.
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Systems of equations with absolute value
In systems like |x| + y = 2 and x - y = 1, solve by considering cases for the absolute value and substituting into the other equations.
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Absolute value word problems
Absolute value in word problems often represents differences or distances, such as the deviation from a target value.
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Interval notation for |x| < 5
The solution to |x| < 5 in interval notation is (-5, 5), indicating all x between -5 and 5, not including the endpoints.
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Positive values inside absolute value
If the expression inside an absolute value is positive, the absolute value equals the expression itself.
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Negative values inside absolute value
If the expression inside an absolute value is negative, the absolute value equals the negative of the expression.
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Absolute value and squares
The square of a number equals the square of its absolute value, since (|x|)^2 = x^2.
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Example: Solve |2x - 3| = 1
To solve |2x - 3| = 1, the solutions are 2x - 3 = 1 or 2x - 3 = -1, giving x = 2 or x = 1.
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Example: Solve |x - 4| > 3
For |x - 4| > 3, the solutions are x - 4 > 3 or x - 4 < -3, so x > 7 or x < 1.
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Example: Solve |x + 1| ≤ 2
For |x + 1| ≤ 2, the solution is -2 ≤ x + 1 ≤ 2, which simplifies to -3 ≤ x ≤ 1.
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Example: Graph y = |x - 2|
The graph of y = |x - 2| is a V-shape shifted right by 2 units, with the vertex at (2, 0).
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Example: Nested absolute value | |x| - 1 |
For | |x| - 1 |, first find |x|, then subtract 1 and take the absolute value, resulting in different expressions based on x.
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Example: |x^2 - 4| = 0
Solving |x^2 - 4| = 0 means x^2 - 4 = 0, so x^2 = 4, giving x = 2 or x = -2.
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Example: Minimize |x - 5| + |x - 1|
The minimum of |x - 5| + |x - 1| occurs at x between 1 and 5, specifically at any point in [1, 5], with the value 4.
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Example: |3x - 6| + 2 = 4
Solving |3x - 6| + 2 = 4 simplifies to |3x - 6| = 2, so 3x - 6 = 2 or 3x - 6 = -2, giving x = 8/3 or x = 4/3.
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Example: Word problem with absolute value
If a number is 5 units from 3, it satisfies |x - 3| = 5, so x = 8 or x = -2.
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Absolute value in inequalities with variables
In inequalities like |x - a| < b, the variable x must lie within b units of a on the number line.
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Common trap: Absolute value of negative numbers
A frequent mistake is thinking | -x | equals -x; instead, it equals x if x is positive.
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Absolute value and equality
Two expressions are equal in absolute value if they are either equal or negatives of each other, like |a| = |b| implies a = b or a = -b.
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Using absolute value for differences
Absolute value calculates the positive difference between two numbers, such as |a - b| giving the larger minus the smaller.
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Absolute value in optimization
In GMAT problems, absolute value helps find maximum or minimum distances, like minimizing |x - a| subject to constraints.
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Solving |ax + b| = |cx + d|
To solve |ax + b| = |cx + d|, square both sides or consider cases: ax + b = cx + d or ax + b = -(cx + d).
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Absolute value with multiple variables
Expressions like |x + y| require considering combinations of signs for x and y when solving equations.
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Example: |x| + |x - 2| = 2
Solving |x| + |x - 2| = 2 involves cases: for x ≥ 2, it becomes x + (x - 2) = 2, so x = 2; check other intervals.
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Absolute value and even functions
The function f(x) = |x| is even, meaning f(-x) = f(x), which can help in symmetry problems.
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Common trap: Extraneous solutions
When solving absolute value equations, always verify solutions to avoid extraneous ones from incorrect case handling.
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Absolute value in rational expressions
In expressions like 1 / |x|, the absolute value ensures the denominator is positive, affecting the sign of the overall fraction.
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Example: |2x - 1| > |x + 3|
To solve |2x - 1| > |x + 3|, consider cases or square both sides carefully, leading to intervals like x > 4 or -1 < x < 1.