Inequalities
57 flashcards covering Inequalities for the SAT Math section.
Inequalities are mathematical statements that compare two expressions using symbols like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). They show that one value is not exactly equal to another but falls within a certain range, such as x > 3 meaning x could be 4, 5, or any larger number. This concept is essential for solving real-world problems, like determining budgets or speeds, and it builds a foundation for more advanced math.
On the SAT Math section, inequalities appear in algebra questions, such as solving linear inequalities, graphing them on the coordinate plane, or working with systems and absolute values. Common traps include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, or misinterpreting compound inequalities like "and" or "or" statements. Focus on practicing manipulation rules, translating word problems into inequalities, and checking solutions carefully to avoid these errors.
Always double-check inequality signs after multiplying or dividing by negatives.
Terms (57)
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Inequality
A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, indicating one is less than, greater than, less than or equal to, or greater than or equal to the other.
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Linear inequality
An inequality that involves a linear expression, typically of the form ax + b < c or similar, where a, b, and c are constants and x is the variable.
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Solving a linear inequality
The process of finding all values of the variable that make the inequality true, similar to solving equations but reversing the inequality sign when multiplying or dividing by a negative number.
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Addition property of inequality
A rule that allows adding the same number to both sides of an inequality without changing the direction of the inequality sign.
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Subtraction property of inequality
A rule that allows subtracting the same number from both sides of an inequality without changing the direction of the inequality sign.
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Multiplication property of inequality
A rule that allows multiplying both sides of an inequality by the same positive number without changing the sign, but requires reversing the sign if multiplying by a negative number.
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Division property of inequality
A rule that allows dividing both sides of an inequality by the same positive number without changing the sign, but requires reversing the sign if dividing by a negative number.
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Compound inequality
An inequality that combines two or more inequalities, such as a < x < b, which means x is greater than a and less than b.
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And inequality
A compound inequality where both conditions must be true simultaneously, often written as a ≤ x ≤ b.
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Or inequality
A compound inequality where at least one condition must be true, such as x < a or x > b.
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Absolute value inequality
An inequality involving the absolute value of an expression, like |x - 2| < 3, which means the distance between x and 2 is less than 3.
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Solving absolute value inequalities
For |expression| < number, it means the expression is between -number and number; for |expression| > number, it means the expression is less than -number or greater than number.
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Quadratic inequality
An inequality involving a quadratic expression, such as x² + 2x - 3 > 0, which requires finding the values of x that make the quadratic positive or negative.
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System of inequalities
A set of two or more inequalities that must be solved together, often by finding the overlapping region on a graph.
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Graphing linear inequalities
The process of plotting the boundary line and shading the region that satisfies the inequality, such as shading above the line for y > mx + b.
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Boundary line
The line that represents the equality part of an inequality, like y = mx + b, which is drawn as solid for ≤ or ≥ and dashed for < or >.
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Test point method
A technique for graphing inequalities where you pick a point not on the boundary line and check if it satisfies the inequality to determine which side to shade.
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Shading region for inequalities
The area on a graph that represents all points satisfying the inequality, such as the region above a line for y > something.
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Inequality with fractions
An inequality involving fractions, solved by multiplying through by the least common denominator while considering the sign of the multiplier.
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Flipping the inequality sign
Occurs when multiplying or dividing both sides of an inequality by a negative number, requiring the inequality symbol to reverse direction.
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Common trap: dividing by a variable
A mistake in inequalities where dividing both sides by a variable that could be positive or negative leads to incorrect results without considering the sign.
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Difference between inequalities and equations
Equations have exact solutions where expressions are equal, while inequalities have a range of solutions where one expression is greater or less than another.
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Word problems with inequalities
Problems that translate real-world scenarios into inequalities, such as setting up x > 10 to mean something costs more than 10 dollars.
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Interval notation
A way to express the solution to an inequality using parentheses and brackets, like (2, 5] for numbers greater than 2 and less than or equal to 5.
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Number line representation
A visual way to show inequality solutions by marking points on a line, using open circles for strict inequalities and closed circles for inclusive ones.
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Solving inequalities with variables on both sides
The process of isolating the variable by moving terms to one side, similar to equations, but watching for sign flips if multiplying or dividing by negatives.
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Rational inequality
An inequality involving rational expressions, like (x+1)/(x-2) > 0, solved by finding critical points and testing intervals.
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Sign chart method
A technique for solving inequalities by creating a chart to track the sign of each factor in different intervals determined by the roots.
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Critical points in inequalities
The values of the variable that make the expression zero or undefined, which divide the number line into intervals for testing inequalities.
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Positive and negative intervals
Regions on the number line where the expression is positive or negative, determined by testing points in each interval after finding critical points.
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Extraneous solutions in inequalities
Values that arise from solving but do not satisfy the original inequality, often checked by plugging back in, especially after operations like squaring.
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Union of inequalities
The combination of solutions from two or more inequalities where the result includes all values that satisfy any of them.
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Intersection of inequalities
The set of values that satisfy all inequalities in a system simultaneously.
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Feasibility region
The overlapping area on a graph that satisfies all inequalities in a system, often a polygon for linear systems.
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Minimum and maximum values in inequalities
The smallest and largest values within the solution set, found by evaluating endpoints or vertices in a feasibility region.
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Slope in inequalities
The measure of steepness in the boundary line of a linear inequality, affecting how the shaded region is oriented.
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Parallel lines in inequalities
Lines with the same slope that may form boundaries in systems, creating unbounded regions if they don't intersect.
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Common mistake: squaring both sides
An error in solving inequalities like |x| > 2 by squaring, which can introduce extraneous solutions or change the inequality.
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Inequality applications: age problems
Problems where inequalities model age relationships, such as x > y + 5 meaning one person is more than 5 years older than another.
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Inequality applications: mixture problems
Scenarios involving mixtures where inequalities ensure concentrations or amounts meet certain criteria, like at least 50% solution.
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Inequality applications: distance-rate-time
Problems using inequalities to compare distances, rates, and times, such as a trip taking less than a certain number of hours.
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Optimization problems with inequalities
Problems that use inequalities to find the best value within constraints, like maximizing profit subject to resource limits.
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Factoring quadratics for inequalities
A method to solve quadratic inequalities by factoring the expression and determining where it is positive or negative.
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Completing the square for inequalities
A technique to rewrite a quadratic inequality in vertex form to identify the intervals where it holds true.
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Discriminant in quadratic inequalities
The value b² - 4ac in a quadratic, which helps determine the number of roots and thus the intervals for the inequality.
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Asymptotes in rational inequalities
Vertical lines where the expression is undefined, serving as critical points that divide the number line for solving.
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Domain restrictions in inequalities
Values of the variable that make denominators zero in rational inequalities, excluding them from the solution set.
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Exponential inequalities
Inequalities involving exponential expressions, solved by considering the properties of exponents and testing intervals.
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Logarithmic inequalities
Inequalities with logarithms, solved by using log properties and ensuring the argument is positive.
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Circles and inequalities
Inequalities representing regions inside or outside a circle, like x² + y² < r² for points inside.
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Perpendicular lines in inequalities
Lines with negative reciprocal slopes that may intersect to form boundaries in systems of inequalities.
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Inequalities in functions
Using inequalities to describe the range or domain of functions, such as where a function is increasing or decreasing.
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Strategy for solving compound inequalities
Isolate the variable in the middle by performing the same operation on all parts, ensuring to flip signs if necessary.
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Strategy for graphing systems of inequalities
Graph each inequality separately and find the overlapping shaded region that satisfies all conditions.
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Common trap: assuming equality
A mistake where solutions treat inequalities as equations, forgetting to include or exclude boundary points based on the sign.
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Inequality with multiple variables
An inequality involving more than one variable, often solved graphically to find the feasible region.
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Example: Solve 2x + 3 > 7
Subtract 3 from both sides to get 2x > 4, then divide by 2 to get x > 2, so the solution is all x greater than 2.
For x = 3, 2(3) + 3 = 9, which is greater than 7.