Final answer:
To solve the inequality ||x - 5| + 2| ≤ 3 for x, we consider different cases depending on the signs of |x - 5| and |. The solutions are x ≤ 1, x ≥ 7, or 2 ≤ x ≤ 4.
Step-by-step explanation:
To solve the inequality ||x - 5| + 2| ≤ 3 for x, we need to consider the different cases depending on the signs of |x - 5| and |. First, let's consider the case when both |x - 5| and | are positive. In this case, the inequality becomes (x - 5) + () ≤ 3. Simplifying this, we get x + - 5 + () ≤ 3. Now, let's consider the case when both |x - 5| and | are negative. In this case, the inequality becomes -(x - 5) + () ≤ 3. Simplifying this, we get -x + 5 + () ≤ 3.
Now, let's solve each of these inequalities separately:
Case 1: x + - 5 + () ≤ 3
To simplify this inequality further, we need to consider different subcases depending on the values of x:
Subcase 1.1: x - 5 ≥ 0 and ≥ 0
In this subcase, the inequality becomes x - 5 + ≤ 3. Combining like terms, we get x ≤ 1.
Subcase 1.2: x - 5 < 0 and < 0
In this subcase, the inequality becomes -x + 5 + ≤ 3. Combining like terms, we get x ≥ 7.
Subcase 1.3: x - 5 < 0 and ≥ 0
In this subcase, the inequality becomes -x + 5 + () ≤ 3. Combining like terms, we get x ≥ 2.
Subcase 1.4: x - 5 ≥ 0 and < 0
In this subcase, the inequality becomes x - 5 + () ≤ 3. Combining like terms, we get x ≤ 4.
Case 2: -x + 5 + () ≤ 3
To simplify this inequality further, we need to consider different subcases depending on the values of x:
Subcase 2.1: x - 5 ≥ 0 and ≥ 0
In this subcase, the inequality becomes -x + 5 + ≤ 3. Combining like terms, we get x ≤ 1.
Subcase 2.2: x - 5 < 0 and < 0
In this subcase, the inequality becomes x - 5 + ≤ 3. Combining like terms, we get x ≥ 7.
Subcase 2.3: x - 5 < 0 and ≥ 0
In this subcase, the inequality becomes -x + 5 + () ≤ 3. Combining like terms, we get x ≥ 2.
Subcase 2.4: x - 5 ≥ 0 and < 0
In this subcase, the inequality becomes -x + 5 + () ≤ 3. Combining like terms, we get x ≤ 4.
Now, let's put all the solutions together:
x ≤ 1 OR x ≥ 7 OR 2 ≤ x ≤ 4.