To solve an inequality, you can follow these steps:
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Get all variables on one side and constants on the other: Use inverse operations to move terms around. This means if you have a variable term (like \(x\)) on one side and a constant (like \(5\)) on the other, you can move them using addition or subtraction.
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Isolate the variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable. If you have an expression like \(2x + 3 < 7\), you'd first subtract \(3\) from both sides, and then divide by \(2\).
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Handle the inequality: Unlike equations, you should be cautious when multiplying or dividing both sides of an inequality. If you multiply or divide by a negative number, you must reverse the direction of the inequality.
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Consider the boundary case: If you change the inequality to an equation (e.g., \(2x + 3 = 7\)), you can find critical points that help identify where the solution region is.
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Test intervals: After finding boundary points, you can test intervals to see where the inequality holds true.
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Write the solution: Finally, express the solution in interval notation or using set notation as appropriate.
Here's an example to illustrate the process:
Example: Solve the inequality \(2x + 3 < 7\).
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Subtract \(3\) from both sides: \[ 2x < 4 \]
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Divide both sides by \(2\): \[ x < 2 \]
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The solution set is all numbers less than \(2\), which can be expressed in interval notation as: \[ (-\infty, 2) \]
By following these steps, you can systematically solve inequalities and understand the solution set!
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