Absolute value
46 flashcards covering Absolute value for the ACT Math section.
Absolute value is a way to measure the distance of a number from zero on the number line, without considering direction. For instance, the absolute value of 7 is 7, and the absolute value of -7 is also 7, making it always a non-negative number. This concept helps in understanding magnitudes in math, such as in distances, errors, or real-world scenarios like temperatures, and it's a building block for more complex topics like equations and inequalities.
On the ACT Math section, absolute value appears in problems involving solving equations (e.g., |x + 2| = 4, which yields two solutions), inequalities (e.g., |x - 3| < 5), and word problems requiring interpretation of distances. Common traps include overlooking the two possible cases for equations or reversing inequality signs incorrectly when manipulating expressions. Focus on mastering the basic rules, practicing graphing absolute value functions, and double-checking for extraneous solutions to handle these questions efficiently, as they often test algebraic accuracy under time pressure.
Remember to always consider both positive and negative roots when solving absolute value equations.
Terms (46)
- 01
Definition of absolute value
The absolute value of a number is its distance from zero on the number line, always a non-negative value.
- 02
Absolute value of a positive number
For a positive number a, the absolute value |a| is a itself, since it is already positive.
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Absolute value of a negative number
For a negative number a, the absolute value |a| is -a, making it positive.
- 04
Absolute value of zero
The absolute value of zero is zero, as it is neither positive nor negative.
- 05
Property: | -a | = | a |
The absolute value of the negative of a number a is the same as the absolute value of a, because distance is unaffected by direction.
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Property: | a b | = | a | | b |
The absolute value of the product of two numbers a and b equals the product of their absolute values.
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Triangle inequality: | a + b | ≤ | a | + | b |
For any numbers a and b, the absolute value of their sum is less than or equal to the sum of their absolute values.
- 08
Solving | x | = c where c > 0
To solve | x | = c, where c is positive, the solutions are x = c or x = -c.
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No solution for | x | = c where c < 0
If c is negative, the equation | x | = c has no solution because absolute values are always non-negative.
- 10
Solving | x | > c where c > 0
The inequality | x | > c means x > c or x < -c.
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Solving | x | < c where c > 0
The inequality | x | < c means -c < x < c.
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Two-case method for absolute value equations
For an equation like | ax + b | = c, consider two cases: ax + b = c and ax + b = -c, then solve each and check for extraneous solutions.
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Graph of y = | x |
The graph of y = | x | is a V-shaped line with its vertex at the origin (0,0), rising linearly on both sides.
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Vertex of y = | x |
The vertex of the graph y = | x | is at (0,0), the lowest point where the two linear pieces meet.
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General form: y = a | x - h | + k
This represents an absolute value function shifted h units horizontally and k units vertically, with a affecting the slope and direction.
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Effect of a in y = a | x - h | + k
If a > 0, the graph opens upwards; if a < 0, it opens downwards, and |a| determines the steepness.
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Effect of h in y = a | x - h | + k
The value h shifts the graph horizontally: positive h moves it right, negative h moves it left.
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Effect of k in y = a | x - h | + k
The value k shifts the graph vertically: positive k moves it up, negative k moves it down.
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Domain of absolute value functions
The domain of any absolute value function like y = a | x - h | + k is all real numbers, as there are no restrictions on x.
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Range of y = a | x - h | + k when a > 0
For a > 0, the range is y ≥ k, starting from the vertex and going upwards.
- 21
Absolute value as distance
Absolute value represents the distance between two points on the number line, for example, | b - a | is the distance between a and b.
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Distance between two points
The distance between points a and b on the number line is | b - a |, which is always positive regardless of order.
- 23
Word problems with absolute value and distance
In word problems, absolute value often models distances, such as 'a point is 5 units from 3' translating to | x - 3 | = 5.
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Common mistake: Ignoring the negative case
When solving absolute value equations, forgetting to consider the negative case can lead to missing solutions.
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Absolute value of fractions
The absolute value of a fraction is the fraction with its numerator's absolute value over its denominator's absolute value, always positive.
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Absolute value of sums
For numbers a and b, | a + b | is the absolute value of their sum, which may not equal | a | + | b |.
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Absolute value of differences
The absolute value of a - b, or | a - b |, gives the positive difference between a and b.
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Nested absolute values
Expressions like | | x | - 2 | require evaluating the inner absolute value first, then the outer one.
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Piecewise definition of absolute value
The function | x | can be defined piecewise as x if x ≥ 0, and -x if x < 0.
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Writing | x | as a piecewise function
To write | x | piecewise: if x ≥ 0, y = x; if x < 0, y = -x.
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Symmetry of absolute value graphs
Graphs of absolute value functions are symmetric about their vertex line, reflecting over the line x = h.
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Minimum value of y = a | x - h | + k
For a > 0, the minimum value is k at the vertex; for a < 0, there is no minimum, only a maximum.
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Graph opens upwards when a > 0
In y = a | x - h | + k, if a is positive, the V-shape points upwards from the vertex.
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Graph opens downwards when a < 0
In y = a | x - h | + k, if a is negative, the V-shape points downwards from the vertex.
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Solving | ax + b | = c
Solve by considering ax + b = c or ax + b = -c, then verify solutions in the original equation.
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Solving | ax + b | > c
This inequality holds when ax + b > c or ax + b < -c.
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Solving | ax + b | < c
This inequality holds when -c < ax + b < c.
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Absolute value in inequalities with 'and'
For | expression | < c, it combines two inequalities with 'and': expression > -c and expression < c.
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Absolute value for tolerances
In contexts like errors, | measured - actual | < tolerance means the measured value is within the tolerance of the actual.
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Simplifying | x^2 |
Since x^2 is always non-negative, | x^2 | simplifies to x^2.
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Absolute value of even functions
Absolute value functions like | f(x) | are even if f(x) is even, meaning they are symmetric about the y-axis.
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Factoring with absolute values
Expressions like | x | (x + 1) can be factored, but remember absolute value affects the sign inside.
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Common trap: Squaring both sides
Squaring absolute value equations can introduce extraneous solutions, so always check answers.
- 44
Graphs of absolute value inequalities
For inequalities like y > | x |, graph the absolute value and shade the region above it.
- 45
Shading for | x | < 2
The solution | x | < 2 means shading the region between x = -2 and x = 2 on the number line.
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Boundary lines in absolute value inequalities
In graphing, the boundary is the absolute value graph itself, solid for ≤ or ≥, dashed for < or >.