Problem 282 Solve the inequality \(|5-2 \mat... [FREE SOLUTION]

Chapter 12: Problem 282

Solve the inequality \(|5-2 \mathrm{x}|>3\).

Short Answer

Expert verified

The inequality \(|5-2x| > 3\) is true for values of \(x\) such that \(x < 1 \textrm{ or } x > 4\).

Step by step solution

1. Rewrite the inequality without absolute value

The absolute value of a number \(x\) is the distance from \(x\) to 0 on the number line. We can rewrite the absolute value inequality without the absolute value by considering two cases: one where the expression inside the absolute value is positive and one where the expression inside the absolute value is negative. Case 1: \(5 - 2x \geq 0\) This case corresponds to the situation where the expression inside the absolute value is non-negative. Case 2: \(-(5 - 2x) \geq 0\) This case corresponds to the situation where the expression inside the absolute value is negative. Now we need to solve each case separately and then combine the solutions.

2. Solve case 1

For this case, the inequality is given by: \[(5 - 2x) > 3\] Subtract 3 from both sides: \[2x < 5-3\] Simplify: \[2x < 2\] Divide by 2: \[x < 1\]

3. Solve case 2

For this case, the inequality is given by: \[-(5 - 2x) > 3\] Distribute the negative sign to both terms inside the parentheses, which gives: \[-5 + 2x > 3\] Add 5 to both sides: \[2x > 3+5\] Simplify: \[2x > 8\] Divide by 2: \[x > 4\]

4. Combine the solution sets

Now that we have solutions for both cases, we can combine them to form the full solution set. Since both cases are valid, we use the union of the solution sets. For case 1, the solution set is \(x < 1\), and for case 2, the solution set is \(x > 4\). So the combined solution set is given by \(x < 1 \textrm{ or } x > 4\). The inequality \(|5-2x| > 3\) is true for any value of \(x\) that satisfies the combined solution set, namely, \(x < 1 \textrm{ or } x > 4\).

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91影视

Solve the inequality \(|5-2 \mathrm{x}|>3\).

Short Answer

Step by step solution

1. Rewrite the inequality without absolute value

2. Solve case 1

3. Solve case 2

4. Combine the solution sets

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