How to Solve Inequalities With Interval Notation

Your desire to learn mathematics is greater than your desire to play video games. That's an inequality. And a true statement, right?
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If you were given the equation x + 2 = 4, it probably wouldn’t take you long to figure out that x = 2. No other number will substitute for x and make that a true statement. If the equation were x^2 + 2 = 4, you would have two answers √2 and -√2. But if you were given the inequality x + 2 < 4, there are an infinite number of solutions. To describe this infinite set of solutions, you would use interval notation, and provide the boundaries of the range of numbers constituting a solution to this inequality.

    Use the same procedures you use when solving equations to isolate your unknown variable. You can add or subtract the same number on both sides of the inequality, just like with an equation. In the example x + 2 < 4 you could subtract two from both the left and right side of the inequality and get x < 2.

    Multiply or divide both sides by the same positive number just as you would in an equation. If 2x + 5 < 7, first you would subtract five from each side to get 2x < 2. Then divide both sides by 2 to get x < 1.

    Switch the inequality if you multiply or divide by a negative number. If you were given 10 - 3x > -5, first subtract 10 from both sides to get -3x > -15. Then divide both sides by -3, leaving x on the left side of the inequality, and 5 on the right. But you’d need to switch the direction of the inequality: x < 5

    Use factoring techniques to find the solution set of a polynomial inequality. Suppose you were given x^2 - x < 6. Set your right side equal to zero, as you would when solving a polynomial equation. Do this by subtracting 6 from both sides. Because this is subtraction, the inequality sign does not change. x^2 - x - 6 < 0. Now factor the left side: (x+2) (x-3) < 0. This will be a true statement when either (x+2) or (x-3) is negative, but not both, because the product of two negative numbers is a positive number. Only when x is > -2 but < 3 is this statement true.

    Use interval notation to express the range of numbers making your inequality a true statement. The solution set describing all numbers between -2 and 3 is expressed as: (-2,3). For the inequality x + 2 < 4, the solution set includes all numbers less than 2. So your solution ranges from negative infinity up to (but not including) 2 and would be written as (-inf, 2).

    Use brackets instead of parentheses to indicate that either or both of the numbers serving as boundaries for the range of your solution set are included in the solution set. So if x + 2 is less than or equal to 4, 2 would be a solution to the inequality, in addition to all the numbers less than 2. The solution to this would be written as: (-inf, 2]. If the solution set were all numbers between -2 and 3, including -2 and 3, the solution set would be written as: [-2,3].


About the Author

Andrew Breslin has been writing professionally since 1994. His articles and op-ed pieces have appeared in the "South Florida Sun Sentinel," "St Paul Pioneer Press," "Detroit Free Press," "Charlotte Observer," “Good Medicine,” and others. He studied molecular biology at Westchester University and frequently writes about science and mathematics.

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