Inequalities
55 flashcards covering Inequalities for the ACT Math section.
Inequalities are mathematical statements that compare two values, showing that one is greater than, less than, or not equal to the other. For example, an inequality like x > 3 means x must be any number larger than 3. They help us describe ranges and boundaries, which are useful in everyday problems, such as budgeting or measuring distances. Mastering inequalities builds a strong foundation for more advanced math concepts.
On the ACT Math section, inequalities often appear in questions involving solving linear or absolute value inequalities, graphing them on number lines or coordinate planes, and using them in word problems with constraints like maximum or minimum values. Common traps include forgetting to flip the inequality sign when multiplying or dividing by a negative number, or overlooking multiple solution intervals. Focus on practicing accurate manipulation and interpreting results to avoid errors.
Remember to always test your solutions by plugging in values.
Terms (55)
- 01
Inequality
An inequality is a mathematical statement that compares two expressions using symbols like >, <, ≥, or ≤, indicating that one is greater than, less than, greater than or equal to, or less than or equal to the other.
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Linear inequality
A linear inequality is an inequality that involves a linear expression, such as 2x + 3 > 5, and can be solved by isolating the variable on one side using algebraic operations.
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Solving a linear inequality
Solving a linear inequality means finding the values of the variable that make the inequality true, similar to solving equations but remembering to reverse the inequality sign if multiplying or dividing by a negative number.
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Graphing inequalities on a number line
Graphing inequalities on a number line involves plotting the solution set, using an open circle for strict inequalities like > or < and a closed circle for ≥ or ≤, then shading the appropriate direction.
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Inequality symbols
Inequality symbols are > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to), used to compare two values.
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Greater than symbol
The greater than symbol (>) indicates that the value on the left is larger than the value on the right, such as 5 > 3.
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Less than symbol
The less than symbol (<) indicates that the value on the left is smaller than the value on the right, such as 2 < 4.
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Greater than or equal to
The symbol ≥ means the value on the left is greater than or equal to the value on the right, including equality, such as x ≥ 5.
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Less than or equal to
The symbol ≤ means the value on the left is less than or equal to the value on the right, including equality, such as y ≤ 10.
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Compound inequality
A compound inequality combines two inequalities with 'and' or 'or', such as 2 < x < 5, which means x is greater than 2 and less than 5.
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And inequality
An 'and' inequality requires both conditions to be true simultaneously, like x > 3 and x < 7, which is written as 3 < x < 7.
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Or inequality
An 'or' inequality is true if at least one condition is met, such as x > 4 or x < 2, resulting in a solution set that combines the intervals.
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Interval notation
Interval notation expresses solution sets of inequalities using parentheses for open ends and brackets for closed ends, such as (2, 5) for values greater than 2 and less than 5.
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Absolute value inequality
An absolute value inequality involves the absolute value of an expression, like |x - 3| > 2, and requires considering cases where the expression inside is positive or negative.
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Solving |x| > a
To solve |x| > a where a > 0, the solution is x < -a or x > a, meaning x is more than a units away from zero on the number line.
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Solving |x| < a
To solve |x| < a where a > 0, the solution is -a < x < a, meaning x is less than a units away from zero.
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Quadratic inequality
A quadratic inequality involves a quadratic expression, like x^2 - 4 > 0, and is solved by finding the roots of the corresponding equation and testing intervals.
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System of inequalities
A system of inequalities consists of two or more inequalities that must be solved together, with the solution being the set of points that satisfy all inequalities simultaneously.
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Feasible region
The feasible region is the area on a graph that satisfies all inequalities in a system, often shaded to represent possible solutions.
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Addition property of inequality
The addition property of inequality states that adding the same number to both sides of an inequality preserves the inequality, such as if x > y, then x + 5 > y + 5.
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Subtraction property of inequality
The subtraction property of inequality states that subtracting the same number from both sides preserves the inequality, like if x > y, then x - 3 > y - 3.
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Multiplication property of inequality
The multiplication property of inequality states that multiplying both sides by a positive number preserves the inequality, but reverses it if multiplying by a negative number.
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Division property of inequality
The division property of inequality states that dividing both sides by a positive number preserves the inequality, but reverses it if dividing by a negative number.
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Reversing inequality sign
Reversing the inequality sign is necessary when multiplying or dividing both sides by a negative number to maintain the truth of the inequality.
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Inequalities with fractions
Inequalities with fractions can be solved by multiplying both sides by the reciprocal of the denominator, remembering to reverse the sign if the multiplier is negative.
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Inequalities with variables on both sides
These inequalities require moving all variable terms to one side and constants to the other, then solving as usual, such as 2x + 3 > x - 1.
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Word problems involving inequalities
Word problems involving inequalities translate real-world scenarios into inequality statements, then solve to find ranges of possible values, like 'at least' or 'no more than'.
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Boundary line in inequalities
The boundary line in a linear inequality is the line where the inequality becomes an equality, such as y = 2x + 1 for y > 2x + 1, and it is dashed if the inequality is strict.
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Shading regions for inequalities
Shading regions for inequalities on a graph indicates the area where the inequality holds true, such as shading above the line for y > mx + b.
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Test point method for inequalities
The test point method for inequalities involves picking a point not on the boundary line and substituting it into the inequality to determine which side to shade.
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Rational inequalities
Rational inequalities involve fractions with variables in the denominator, solved by finding critical points and testing intervals while considering where the expression is undefined.
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Inequalities with exponents
Inequalities with exponents, like 2^x > 8, are solved by recognizing equivalent bases or taking logarithms, then isolating the variable.
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Common mistake: forgetting to flip sign
A common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number, which leads to incorrect solutions.
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Union of intervals
The union of intervals combines separate solution sets from inequalities, such as x < -2 or x > 3, written as (-∞, -2) ∪ (3, ∞).
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Intersection of intervals
The intersection of intervals represents values that satisfy multiple inequalities simultaneously, like the overlap of (1, 5) and (3, 7), which is (3, 5).
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Infinite solutions in inequalities
Infinite solutions in inequalities occur when the solution set is an entire interval or the whole number line, such as x > -∞, meaning all real numbers.
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No solution in inequalities
No solution in inequalities happens when no values satisfy the condition, such as x > 5 and x < 3, which is impossible.
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Strict inequality
A strict inequality uses > or < and does not include equality, so solutions are open intervals on a graph.
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Non-strict inequality
A non-strict inequality uses ≥ or ≤ and includes equality, so solutions include the boundary points on a graph.
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Example: 2x + 3 > 7
For the inequality 2x + 3 > 7, subtract 3 from both sides to get 2x > 4, then divide by 2 to find x > 2.
Solution: x > 2, graphed as an open circle at 2 shading right.
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Example: -4x ≤ 12
For -4x ≤ 12, divide both sides by -4 and reverse the inequality to get x ≥ -3.
Solution: x ≥ -3, graphed as a closed circle at -3 shading right.
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Graphing y > 2x + 1
Graphing y > 2x + 1 involves drawing the line y = 2x + 1 as dashed and shading the region above it where the inequality holds.
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Solving x^2 - 4 > 0
To solve x^2 - 4 > 0, factor to (x - 2)(x + 2) > 0, find roots at x = 2 and x = -2, then test intervals to get x < -2 or x > 2.
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System: x + y > 5 and x - y < 3
For the system x + y > 5 and x - y < 3, graph both inequalities and find the overlapping shaded region as the solution.
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Inequality in functions
Inequality in functions compares function outputs, such as f(x) > g(x), by analyzing their graphs or solving the inequality f(x) - g(x) > 0.
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Domain restrictions in inequalities
Domain restrictions in inequalities exclude values that make denominators zero or cause other undefined expressions, such as in rational inequalities.
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Extraneous solutions in inequalities
Extraneous solutions in inequalities are values that satisfy the inequality but are invalid due to domain restrictions, like making a denominator zero.
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Clearing fractions in inequalities
Clearing fractions in inequalities involves multiplying both sides by the least common denominator, reversing the sign if the LCD is negative.
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Strategy for solving absolute value inequalities
The strategy for solving absolute value inequalities is to consider the definition of absolute value and split into cases or use the rules for |x| > a and |x| < a.
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Common trap: Assuming equality
A common trap is treating an inequality like an equality, which overlooks the range of solutions and can lead to incorrect graphing or interpretation.
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Inequalities with square roots
Inequalities with square roots require ensuring the expression inside is non-negative and solving while considering the domain, such as √x > 2 implying x > 4.
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Three-part inequality
A three-part inequality, like -1 < x < 3, combines two inequalities and requires all parts to be true simultaneously.
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Shade above or below for y-inequalities
For linear inequalities in two variables, shade above the line if y > mx + b and below if y < mx + b.
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Using a sign chart for polynomials
A sign chart for polynomial inequalities plots roots on a number line and tests the sign of the expression in each interval to determine where it's positive or negative.
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Inequalities in circles
Inequalities in circles, like (x - h)^2 + (y - k)^2 < r^2, represent the interior of a circle and are graphed by shading inside the boundary.