Solving Inequalities Involving |x+2| & |x2 -3ax+2a2|

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In summary, for the first conversation, the task is to solve for x in terms of a for the inequality |x2 - 3ax + 2a2| < |x2 + 3a - a2|, where x is real and a is not equal to 0. For the second conversation, the task is to determine the range of values of a for which the equation |x+2| = ax + 4 has 2 distinct real roots, and to solve the inequality |x+2| < ax + 4.
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1) Solve for x, in terms of a, the inequality |x2 -3ax + 2a2| < |x2 +3a - a2| where x is real . a is not 0.
2ai) By means of a sketch or otherwise, state the range of values of a for which the equation |x+2| = ax + 4 has 2 distinct real roots.
2aii)Solve the inequality |x+2| < ax + 4.
 
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Hello A Level Student and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

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FAQ: Solving Inequalities Involving |x+2| & |x2 -3ax+2a2|

1. What are inequalities involving absolute values?

Inequalities involving absolute values are mathematical statements that contain at least one absolute value expression, such as |x+2| or |x2 - 3ax + 2a2|. These expressions represent the distance of a number from zero, and the inequality states that the value of the expression must be less than or greater than a certain number.

2. How do you solve inequalities involving absolute values?

To solve inequalities involving absolute values, you must first isolate the absolute value expression on one side of the inequality. Then, you must consider two cases: when the value inside the absolute value is positive and when it is negative. For each case, you can remove the absolute value by applying the appropriate inequality symbols and solve for the variable.

3. What is the difference between solving inequalities with one absolute value and two absolute values?

Solving inequalities with one absolute value involves only one absolute value expression, while solving inequalities with two absolute values involves two absolute value expressions. This means that there may be two possible solutions for the variable in the latter case.

4. How do you determine the solution set for inequalities involving absolute values?

The solution set for inequalities involving absolute values is the set of all values of the variable that satisfy the inequality. After solving for the variable, you can graph the solution set on a number line or express it in interval notation.

5. What are some real-life applications of inequalities involving absolute values?

Inequalities involving absolute values are commonly used in real-life situations involving distance, such as calculating the minimum or maximum distance between two points, or determining the acceptable range of values for a measurement. They are also used in optimization problems, where the goal is to minimize or maximize a certain value.

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