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Lesson 114: Solving Inequalities
In this quick and easy lesson, we'll learn how to solve algebraic inequalities.
Solving algebraic inequalities is almost the same as solving algebraic equations, with one special rule to remember. Let's solve the inequality x + 3 > 7. What we're looking for is values of x that will make this inequality true. Unlike algebraic equations that have only one solution, inequalities like this will have an infinite number of solutions.
To solve this inequality, we'll just imagine, for the moment, that the > sign is actually an equals sign. Then we'll solve as usual. Subtracting 3 from each side, we have x = 4. We then have to remember to put back our > sign, so our answer is x > 4. Any value of x that is greater than 4 will satisfy the inequality.
Let's try another one. 2x + 3 ≤ 15. Just like above, we'll start by pretending that the ≤ is an equals sign, and we'll solve the equation like we've learned. First we'll subtract 3 from each side, and then we'll divide each side by 2. We end up with x = 6, and we must remember to put back our original inequality sign. We get x ≤ 6. Any value of x that is less than or equal to 6 will satisfy this inequality.
Here is a special case. Let's say we have -3x > 12. As before, we'll temporarily convert the > into an equals sign. Then we'll divide each side by -3, and get x = -4. But now we must do something special. There is a special rule whenever we multiply or divide by a negative number. We must reverse the original inequality sign. In this case we must reverse the >, and we get x < -4. Test it out. We need numbers that are lower than -4, if we want our product to be higher than 12 when we multiply by -3. Make sure you see that this is the case. Memorize this rule for multiplying and dividing by negative numbers.
You'll have more practice with these types of problems in later lessons. Just make sure that you understand the concepts presented, and that you memorize the special rule for multiplying or dividing by a negative number. Whenever you multiply or divide by a negative number, flip the inequality sign after solving the inequality as usual.
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