Absolute value inequalities
51 flashcards covering Absolute value inequalities for the SAT Math section.
Absolute value inequalities involve comparing the absolute value of an expression to a number, which essentially deals with distances on the number line. For example, the inequality |x - 2| < 3 means that x is less than 3 units away from 2, so x could be between -1 and 5. This concept builds on basic inequalities by requiring you to consider both positive and negative possibilities, making it a key tool for solving real-world problems like determining ranges in science or economics.
On the SAT Math section, absolute value inequalities appear in algebra questions, often as multiple-choice or grid-in problems that test your ability to solve and graph them. Common traps include forgetting to account for two cases (e.g., when the expression inside the absolute value is positive or negative) or mishandling the inequality symbol when multiplying or dividing by a negative number. Focus on practicing step-by-step solutions and interpreting results on a number line to avoid errors and handle questions efficiently.
Remember to always test your solution in the original inequality to catch any mistakes.
Terms (51)
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Absolute value
The absolute value of a number is its distance from zero on the number line, always non-negative, so |x| is x if x ≥ 0 and -x if x < 0.
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Solving |x| = a
To solve |x| = a where a > 0, find the values of x that are a units away from zero, resulting in x = a or x = -a.
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Solving |x| < a
For |x| < a where a > 0, the solution is all x such that the distance of x from zero is less than a, which means -a < x < a.
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Solving |x| > a
For |x| > a where a > 0, the solution is all x such that the distance of x from zero is greater than a, which means x > a or x < -a.
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Solving |x + b| < c
To solve |x + b| < c where c > 0, rewrite it as -c < x + b < c, then subtract b from all parts to get -c - b < x < c - b.
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Solving |x + b| > c
To solve |x + b| > c where c > 0, it means x + b > c or x + b < -c, so x > c - b or x < -c - b.
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Solving |ax| < b
For |ax| < b where a and b are positive, divide the inequality by |a| to get -b/a < x < b/a if a > 0, or adjust accordingly for the sign of a.
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Solving |ax| > b
For |ax| > b where a and b are positive, it means ax > b or ax < -b, so x > b/a or x < -b/a if a > 0.
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Graph of |x| < a
The graph of |x| < a on the number line is an open interval from -a to a, representing all points between -a and a, not including the endpoints.
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Graph of |x| > a
The graph of |x| > a on the number line consists of two rays: one from -∞ to -a and another from a to ∞, excluding the points -a and a.
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Absolute value inequality with equality
Absolute value inequalities can include equality, like |x| ≤ a, which combines |x| < a and |x| = a, so -a ≤ x ≤ a.
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Absolute value inequality without equality
For strict inequalities like |x| ≥ a, it means x ≤ -a or x ≥ a, including the endpoints but extending infinitely in those directions.
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Common trap: Forgetting cases
A common error is solving absolute value inequalities by only considering one case, but you must account for both positive and negative possibilities inside the absolute value.
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When a is zero in |x| < a
If a = 0 in |x| < 0, there are no solutions because no number has a distance less than zero from zero.
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When a is negative in |x| < a
If a < 0 in |x| < a, the inequality has no solution since absolute value is always non-negative and can't be less than a negative number.
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Intersection of absolute value inequalities
Sometimes you need to find the intersection of two absolute value inequalities, which means solving both and taking the overlapping solution sets.
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Union of absolute value inequalities
For inequalities like |x| > a or |x| < b, the solution is the union of the individual solution sets.
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Absolute value in word problems
Absolute value often represents distance in word problems, such as 'the distance between x and 5 is less than 3', which translates to |x - 5| < 3.
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Solving |x - b| ≤ c
This means the distance between x and b is at most c, so b - c ≤ x ≤ b + c.
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Solving |x - b| ≥ c
This indicates the distance between x and b is at least c, so x ≤ b - c or x ≥ b + c.
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Absolute value with fractions
When solving inequalities like |2x + 1| < 3, treat it as -3 < 2x + 1 < 3, then subtract 1 and divide by 2 to get -2 < x < 1.
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Multiplying inequalities with absolute values
If you multiply or divide an absolute value inequality by a negative number, you must reverse the inequality sign, but this is rare in basic forms.
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Absolute value and linear inequalities
Absolute value inequalities can be combined with linear inequalities, requiring you to solve the system by considering the absolute value cases.
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Graphing |x - h| + k < c
This represents points within c units of (h, k) on a graph, forming a region inside a V-shaped boundary.
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Common trap: Misinterpreting the variable
In absolute value inequalities, ensure the variable inside is isolated before applying the rules, as extra terms can change the solution.
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Solving inequalities with two absolute values
For something like |x - 1| + |x - 3| < 2, test intervals based on the points where the expressions inside change sign, like x=1 and x=3.
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Critical points in absolute value inequalities
Critical points are values that make the expression inside the absolute value zero, dividing the number line into intervals to test.
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Testing intervals for |x| inequalities
After identifying critical points, test a point in each interval to see if it satisfies the inequality, then combine the valid intervals.
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Absolute value inequality: No solution
Inequalities like |x| < -1 have no solution because absolute values are always greater than or equal to zero.
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All real numbers solution
For |x| > -1, the solution is all real numbers since every absolute value is greater than a negative number.
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Distance formula in inequalities
Absolute value inequalities can be interpreted using the distance formula, like |x - a| < b means x is within b units of a.
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Rewriting absolute value inequalities
You can rewrite |expression| < k as -k < expression < k, which is useful for more complex expressions.
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Squaring both sides of absolute value
For inequalities like |x| > a, squaring both sides works since both sides are positive, but it's often unnecessary.
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Absolute value in equations vs. inequalities
Unlike equations where |x| = a has two solutions, inequalities like |x| < a have a range, requiring interval notation.
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Interval notation for solutions
Solutions to absolute value inequalities are expressed in interval notation, such as (-a, a) for |x| < a.
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Absolute value and absolute inequalities
Sometimes problems involve inequalities that are always true or false due to absolute values, like |x| ≥ 0 is always true.
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Combining with other functions
Absolute value inequalities might appear with quadratics or other functions, but on the SAT, they are typically linear.
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Example: |2x - 4| < 6
Solve |2x - 4| < 6 by rewriting as -6 < 2x - 4 < 6, adding 4 to get -2 < 2x < 10, then dividing by 2 to get -1 < x < 5.
For x = 0, |2(0) - 4| = 4 < 6, which works.
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Example: |x + 3| > 2
Solve |x + 3| > 2 to get x + 3 > 2 or x + 3 < -2, so x > -1 or x < -5.
For x = 0, |0 + 3| = 3 > 2, which is true.
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Shifting the absolute value graph
Inequalities like |x - h| < k shift the basic |x| graph horizontally by h units and represent a region around that point.
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Vertical shifts in inequalities
Though less common, absolute value expressions like |x| + k in inequalities adjust the boundary vertically.
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Using number lines for verification
After solving, plot the solution on a number line to verify, marking open or closed circles as needed.
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Common error: Dividing by zero
In absolute value inequalities, avoid dividing by an expression that could be zero, as it might exclude solutions.
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Parametric absolute value inequalities
Sometimes variables are parameters, like solving |x - a| < b for different a and b values.
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Absolute value in systems of inequalities
Solve systems by finding the intersection of the solution sets of each inequality.
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Graphical solution for inequalities
On the coordinate plane, absolute value inequalities define regions, like the area between lines for |x| + |y| < 1.
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Maximizing or minimizing with inequalities
Use absolute value inequalities to find maximum or minimum distances in optimization problems.
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Example: |x - 2| + |x - 4| ≥ 3
Critical points are x=2 and x=4; test intervals: for x < 2, it's -x + 2 - x + 4 ≥ 3 simplifies to -2x + 6 ≥ 3, etc.
For x=3, |3-2| + |3-4| = 1 + 1 = 2, which is not ≥ 3.
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Symmetry in absolute value
Absolute value functions are symmetric about the vertex, which helps in understanding inequality solutions.
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Bounded and unbounded solutions
Solutions to |x| < a are bounded, while |x| > a are unbounded.
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Transforming inequalities
You can transform absolute value inequalities by substitution, like letting y = x - b to simplify.