Linear inequalities
51 flashcards covering Linear inequalities for the GMAT Quantitative section.
Linear inequalities are mathematical expressions that compare two values using symbols like greater than (>), less than (<), or not equal to (≠), where the variables form straight-line relationships. For example, an inequality like 2x + 3 > 7 represents a range of values for x that satisfy the condition, rather than a single solution. They are essential for modeling real-world scenarios, such as budgeting constraints or resource allocation, helping you understand boundaries and possibilities in a flexible way.
On the GMAT Quantitative section, linear inequalities often appear in problem-solving and data sufficiency questions, where you might solve for variables, graph regions, or interpret inequalities in word problems. Common traps include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, or overlooking overlapping solution sets in compound inequalities. Focus on mastering techniques like isolating variables and testing critical points to accurately determine feasible regions.
Remember to always double-check inequality directions when performing operations.
Terms (51)
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Linear inequality
A linear inequality is a mathematical statement that compares two expressions using inequality symbols like greater than, less than, greater than or equal to, or less than or equal to, where the expressions are linear, meaning they are of the form ax + b.
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Inequality symbols
Inequality symbols are < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to), used to compare two values and indicate that one is not equal to the other in a specific way.
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Solution set of an inequality
The solution set of an inequality is the collection of all values that make the inequality true when substituted into the equation, often represented as an interval or on a number line.
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Graphing a linear inequality on a number line
Graphing a linear inequality on a number line involves plotting the boundary point and shading the appropriate region based on the inequality symbol, such as shading to the right for greater than.
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Adding or subtracting in inequalities
When adding or subtracting the same value to both sides of an inequality, the inequality symbol remains the same, allowing you to isolate the variable without changing the direction.
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Multiplying or dividing by a positive number
Multiplying or dividing both sides of an inequality by a positive number keeps the inequality symbol unchanged, preserving the original direction of the inequality.
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Multiplying or dividing by a negative number
Multiplying or dividing both sides of an inequality by a negative number requires reversing the inequality symbol to maintain the truth of the statement.
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Compound inequality
A compound inequality combines two or more inequalities using 'and' or 'or', such as x > 2 and x < 5, which means x is between 2 and 5.
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And inequality
An 'and' inequality requires both conditions to be true simultaneously, resulting in an overlapping solution set, like 1 < x and x < 4, which is 1 < x < 4.
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Or inequality
An 'or' inequality is true if at least one condition is met, leading to a combined solution set that includes all possibilities from either side.
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Interval notation
Interval notation expresses the solution set of an inequality using parentheses or brackets, such as (2, 5) for values greater than 2 and less than 5, where parentheses indicate open ends.
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Absolute value inequality
An absolute value inequality involves an expression inside absolute value bars compared to a number, such as |x - 3| < 5, which translates to values within 5 units of 3.
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Systems of linear inequalities
Systems of linear inequalities consist of two or more inequalities that must be solved together, with the solution being the overlapping region that satisfies all conditions.
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Feasible region
The feasible region is the area on a graph where all inequalities in a system are satisfied, often a polygon formed by the boundary lines.
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Boundary line
The boundary line of a linear inequality is the straight line that represents the equality version of the inequality, such as y = 2x + 1 for y > 2x + 1.
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Shading above or below the line
For a linear inequality in two variables, shading above the line indicates greater than or greater than or equal to, while shading below indicates less than or less than or equal to.
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Word problem inequality
A word problem inequality translates real-world scenarios into mathematical inequalities, such as 'more than 10 items' becoming x > 10.
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Break-even point inequality
In business, a break-even point inequality compares costs and revenues, like total cost ≤ total revenue, to find when a company starts making profit.
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Maximum or minimum values
In systems of inequalities, maximum or minimum values occur at the vertices of the feasible region and are found by evaluating the objective function at those points.
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Using test points
Using test points involves picking a point not on the boundary line of an inequality and substituting it to determine which side of the line satisfies the inequality.
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Inequalities with variables on both sides
Solving inequalities with variables on both sides requires moving all variable terms to one side and constants to the other, following the rules for inequalities.
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No solution inequality
An inequality has no solution if the conditions lead to a contradiction, such as x > 5 and x < 3, which cannot both be true.
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All real numbers solution
An inequality has all real numbers as its solution if it simplifies to a statement that is always true, like x > x - 1.
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Transitive property of inequalities
The transitive property states that if a > b and b > c, then a > c, allowing inequalities to be chained together logically.
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Addition property of inequalities
The addition property allows adding the same number to both sides of an inequality without changing the inequality symbol.
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Multiplication property of inequalities
The multiplication property requires that when multiplying or dividing both sides by a negative number, the inequality symbol must be reversed.
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Linear inequality in two variables
A linear inequality in two variables, like 2x + y > 3, represents a region on the coordinate plane rather than a single line.
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Slope in inequalities
The slope of a linear inequality's boundary line indicates the steepness and direction, affecting how the shaded region is oriented on the graph.
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Parallel lines in inequalities
Parallel lines in systems of inequalities never intersect, potentially creating unbounded feasible regions if they don't overlap.
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Determining inequality from word problem
Determining an inequality from a word problem involves identifying keywords like 'at least' for greater than or equal to and setting up the equation accordingly.
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Example: Solving 2x + 3 > 7
To solve 2x + 3 > 7, subtract 3 from both sides to get 2x > 4, then divide by 2 to get x > 2, meaning the solution is all x greater than 2.
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Example: Graphing y < 3x - 2
To graph y < 3x - 2, first draw the line y = 3x - 2, then shade below it since it's less than, and use a dashed line because it's strictly less than.
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Example: System with x + y > 4 and x - y < 2
For the system x + y > 4 and x - y < 2, graph both inequalities and find the overlapping shaded region as the feasible area.
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Common trap: Forgetting to flip the sign
A common trap is forgetting to reverse the inequality symbol when multiplying or dividing both sides by a negative number, which leads to an incorrect solution.
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Strategy for solving compound inequalities
A strategy for solving compound inequalities is to treat 'and' statements as a single inequality and solve step by step, ensuring both parts hold true.
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Inequality with fractions
Solving an inequality with fractions involves multiplying both sides by the least common denominator, remembering to flip the sign if it's negative.
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Nested inequalities
Nested inequalities involve one inequality inside another, requiring you to solve the inner one first before addressing the outer condition.
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Applications in linear programming
In linear programming, inequalities represent constraints, and the goal is to maximize or minimize a linear objective function within the feasible region.
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Profit maximization inequality
A profit maximization inequality sets up constraints like production limits and compares them to revenue goals to find the optimal production level.
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Combining inequalities
Combining inequalities means adding or subtracting them to eliminate variables, similar to systems of equations, while maintaining the inequality rules.
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Example: Solving -4x + 5 ≤ 1
To solve -4x + 5 ≤ 1, subtract 5 from both sides to get -4x ≤ -4, then divide by -4 and flip the sign to get x ≥ 1.
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Example: Feasible region for x + y ≤ 6 and x ≥ 2
For x + y ≤ 6 and x ≥ 2, the feasible region is the area below the line x + y = 6 and to the right of x = 2, forming a polygonal area.
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Strict vs. non-strict inequalities
Strict inequalities use > or < and do not include the boundary, while non-strict ones use ≥ or ≤ and include it, affecting how graphs are drawn.
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Example: Absolute value |x - 4| ≥ 3
For |x - 4| ≥ 3, this means x - 4 ≥ 3 or x - 4 ≤ -3, so x ≥ 7 or x ≤ 1, representing values at least 3 units from 4.
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Inequality chains
Inequality chains link multiple comparisons, like a < b < c, which must all hold true simultaneously using the transitive property.
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Rationalizing inequalities
Rationalizing inequalities involves clearing fractions by multiplying through by the denominator, flipping the sign if the multiplier is negative.
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Unbounded feasible regions
Unbounded feasible regions occur in systems of inequalities when the shaded area extends infinitely, such as in x > 1 and y > 2.
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Cost and revenue inequality
A cost and revenue inequality, like total revenue > total cost, helps determine when a business is profitable by comparing income and expenses.
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Example: Solving 3(x - 2) > 6
To solve 3(x - 2) > 6, divide both sides by 3 to get x - 2 > 2, then add 2 to both sides to get x > 4.
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Strategy for graphing inequalities
A strategy for graphing inequalities is to first graph the equality line, decide if it's solid or dashed, then test a point to shade the correct region.
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Dependent inequalities
Dependent inequalities are those where one inequality's solution affects the other, requiring simultaneous solving to find the valid range.