Absolute value equations
65 flashcards covering Absolute value equations for the SAT Math section.
Absolute value equations involve finding the values of a variable that make an equation true, where the absolute value represents the distance of a number from zero on the number line. For instance, in |x - 4| = 3, the absolute value means x is 3 units away from 4, so x could be 7 or 1. These equations require considering both positive and negative scenarios inside the absolute value, which helps solve for multiple solutions and is useful in real-world contexts like distances or errors.
On the SAT Math section, absolute value equations typically show up in algebra problems, including straightforward equations, inequalities, or word problems involving scenarios like travel distances. Common traps include overlooking the need to check both positive and negative cases, leading to incomplete solutions, or confusing equations with inequalities. Focus on mastering the basic steps: isolate the absolute value, set up equations for each case, and verify your answers to avoid errors.
Remember to always test your solutions in the original equation.
Terms (65)
- 01
Absolute value
The absolute value of a number is its distance from zero on the number line, always non-negative, so |x| equals x if x is positive or zero, and -x if x is negative.
- 02
Solving |x| = k
To solve |x| = k where k is positive, x equals k or x equals -k; if k is zero, x is zero; if k is negative, there is no solution.
- 03
Absolute value equation
An absolute value equation is one that includes an absolute value expression, typically solved by considering the definition of absolute value or by isolating and removing the absolute value.
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Extraneous solution
An extraneous solution is a value that emerges from solving an equation but does not satisfy the original equation, often occurring in absolute value problems if solutions are not checked.
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Compound inequality from absolute value
For an inequality like |x| < k, it becomes -k < x < k; for |x| > k, it becomes x < -k or x > k, representing the solution set.
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|x - a| = b
To solve |x - a| = b where b > 0, x - a = b or x - a = -b, so x = a + b or x = a - b; if b ≤ 0, no solution exists.
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|x + a| = b
Solving |x + a| = b means x + a = b or x + a = -b, so x = b - a or x = -b - a, provided b is positive.
- 08
Solving |ax + b| = c
For |ax + b| = c with c > 0, solve ax + b = c and ax + b = -c separately, then check for extraneous solutions.
- 09
Absolute value inequality
An absolute value inequality compares an absolute value expression to a number, solved by converting to compound inequalities based on whether it's less than or greater than.
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|x| < k
The solution to |x| < k, where k > 0, is -k < x < k, meaning x is within k units of zero on the number line.
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|x| > k
The solution to |x| > k, where k > 0, is x < -k or x > k, meaning x is more than k units away from zero.
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And/or in absolute value solutions
Solutions to absolute value equations often involve 'or' for equalities, like x = a or x = b, while inequalities like |x| < k use 'and' implicitly in the compound form.
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Graph of y = |x|
The graph of y = |x| is a V-shaped figure with its vertex at the origin, consisting of two rays: one with slope 1 for x ≥ 0 and one with slope -1 for x < 0.
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Vertex of absolute value graph
The vertex of a graph like y = |x - h| + k is at the point (h, k), representing the minimum or maximum point of the V-shape.
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Horizontal shift in absolute value graph
In y = |x - h|, h represents the horizontal shift: positive h shifts the graph right by h units, and negative h shifts it left.
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Vertical shift in absolute value graph
In y = |x| + k, k represents the vertical shift: positive k shifts the graph up by k units, and negative k shifts it down.
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Absolute value in distance problems
Absolute value represents distance on the number line, so equations like |x - a| = b mean x is b units away from a.
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Equation for points equidistant from two points
For points equidistant from a and b, the equation is |x - a| = |x - b|, which simplifies to x being the midpoint between a and b.
- 19
Absolute value with variables on both sides
Equations like |ax + b| = |cx + d| are solved by considering cases or squaring both sides, then checking for extraneous solutions.
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|2x - 3| = |x + 1|
To solve |2x - 3| = |x + 1|, set up cases or square both sides: 2x - 3 = x + 1 or 2x - 3 = -(x + 1), leading to x = 4 or x = -2 after verification.
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Solving equations with two absolute values
For equations with two absolute values, like |x| + |x - 2| = 2, consider the points where expressions inside change sign and solve in intervals.
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Absolute value equations with no solution
An absolute value equation has no solution if it implies a negative value equals a positive one, such as |x| = -5.
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Absolute value equations with infinite solutions
Rarely, an absolute value equation like |x| = |x| has infinite solutions, meaning all real numbers satisfy it.
- 24
Rewriting absolute value equations
Absolute value equations can be rewritten using the definition, such as |x| = x if x ≥ 0 and |x| = -x if x < 0, to solve by cases.
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Case method for solving absolute values
The case method involves splitting the equation based on when the expression inside the absolute value is positive or negative, solving each case, and combining valid solutions.
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Squaring both sides for absolute values
For equations like |A| = |B|, squaring both sides gives A² = B², which can be factored as (A - B)(A + B) = 0, but solutions must be checked.
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When to use the definition directly
Use the definition of absolute value directly for simple equations like |x - 2| = 3 by setting x - 2 = 3 or x - 2 = -3.
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Real-world applications on SAT
Absolute value equations model scenarios involving distances, tolerances, or deviations, such as the time until two objects meet or errors within a range.
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Positive value in absolute value
If the expression inside absolute value is positive, |x| simply equals x, as in solving equations where the inside is already non-negative.
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Negative value in absolute value
If the expression inside absolute value is negative, |x| equals -x, turning it positive for equations involving negative inputs.
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Zero in absolute value
The absolute value of zero is zero, so equations like |x| = 0 have the solution x = 0.
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Basic absolute value equation
A basic absolute value equation is something like |x| = 5, solved as x = 5 or x = -5.
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General solution for |x| = k
The general solution for |x| = k is x = k or x = -k when k > 0, x = 0 when k = 0, and no solution when k < 0.
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Solution for |x - a| = b
The solution is x = a + b or x = a - b if b > 0, representing points symmetric around a.
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When b is positive in absolute value
In |expression| = b, if b is positive, there are two potential solutions; if b is zero, one solution; if negative, none.
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When b is zero in absolute value
For |x - a| = 0, the solution is x = a, as the expression inside must be zero.
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When b is negative in absolute value
Equations like |x| = -b have no real solutions because absolute values are always non-negative.
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Solving |x| < 3
The solution to |x| < 3 is -3 < x < 3, an interval on the number line.
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Solving |x| > 3
The solution to |x| > 3 is x < -3 or x > 3, two separate intervals.
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Checking solutions in absolute value equations
Always substitute solutions back into the original equation to ensure they work, as some may be extraneous.
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y = |x + a|
The graph of y = |x + a| is the graph of y = |x| shifted left by a units if a > 0, or right if a < 0.
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y = |x - a| + b
This is y = |x| shifted right by a units and up by b units, with the vertex at (a, b).
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Reflection in absolute value graph
Multiplying inside the absolute value, like y = |-x|, reflects the graph over the y-axis.
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Absolute value in word problems
In word problems, absolute value often represents the difference in quantities, such as |actual - expected| for error margins.
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Common mistake: forgetting to check solutions
A common error is not verifying solutions in absolute value equations, which can lead to incorrect answers due to extraneous values.
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Absolute value with fractions
Equations like |1/2 x - 1| = 2 are solved by multiplying through by the denominator if needed, then applying absolute value rules.
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Absolute value with decimals
For equations like |x - 0.5| = 1.2, solve as x - 0.5 = 1.2 or x - 0.5 = -1.2, resulting in x = 1.7 or x = -0.7.
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Multi-step absolute value equations
These involve additional steps, like distributing or combining like terms, before isolating the absolute value.
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Interpreting absolute value solutions
Solutions to absolute value equations represent values that satisfy the distance condition, often symmetric around a point.
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Number line representation
Absolute value solutions are shown on a number line, with closed circles for equality and open for inequality boundaries.
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Interval notation for solutions
Solutions like -3 < x < 3 are written in interval notation as (-3, 3), while x < -3 or x > 3 is (-∞, -3) ∪ (3, ∞).
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When to use cases for absolute value
Use cases when the expression inside absolute value changes sign, typically at zero, to solve equations accurately.
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Case 1: expression inside positive
In the case method, first assume the inside is positive, solve the equation without absolute value, and check if it fits the assumption.
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Case 2: expression inside negative
Second, assume the inside is negative, rewrite with the negative sign, solve, and verify against the assumption.
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Verifying cases
After solving each case, discard solutions that don't satisfy the original assumption about the sign of the expression.
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Example: Solve |x - 4| = 2
Solving |x - 4| = 2 gives x - 4 = 2 or x - 4 = -2, so x = 6 or x = 2.
On the SAT, this might appear as finding numbers 2 units from 4.
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Example: Solve |2x + 1| < 3
This becomes -3 < 2x + 1 < 3, subtract 1 to get -4 < 2x < 2, divide by 2 to get -2 < x < 1.
Represents x values where 2x + 1 is less than 3 units from zero.
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Example: Solve |x| + |x - 2| = 2
Consider cases: for x ≥ 2, x + (x - 2) = 2 simplifies to 2x - 2 = 2, x = 2; verify and check other intervals.
This equation has solution x = 1, found by testing intervals.
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Strategy for word problems with absolute value
Translate the problem into an absolute value equation by identifying distances or differences, solve it, and interpret the context.
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Absolute value in functions
Absolute value functions like f(x) = |x| are piecewise linear, defined as f(x) = x for x ≥ 0 and f(x) = -x for x < 0.
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Domain of absolute value functions
The domain of absolute value functions, such as y = |x|, is all real numbers, as they are defined everywhere.
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Range of absolute value functions
The range of y = |x| is y ≥ 0, meaning all non-negative values.
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Maximum and minimum values
For y = |x - a| + b, the minimum value is b at x = a, and there is no maximum as it increases indefinitely.
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Absolute value in inequalities with and/or
Inequalities like |x| ≤ k use 'and' for the compound form, while |x| ≥ k uses 'or' for the two intervals.
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Graphing inequalities with absolute value
To graph |x| < k, shade the region between -k and k on the number line; for |x| > k, shade outside that region.