2−|5−2z|≥−12

**Step 1 of 2 :**  

Solve the inequality and express your answer in interval notation. Use decimal form for numerical values.

To solve the inequality \( 2 - |5 - 2z| \geq -12 \), follow these steps:

### Step 1: Isolate the absolute value expression.
First, move the constant term on the left side to the right side of the inequality:

\[
2 - |5 - 2z| \geq -12
\]

Subtract 2 from both sides:

\[
-|5 - 2z| \geq -14
\]

Next, multiply both sides by \(-1\) to remove the negative sign in front of the absolute value. Remember, multiplying or dividing an inequality by a negative number reverses the inequality sign:

\[
|5 - 2z| \leq 14
\]

### Step 2: Write the compound inequality.
The inequality \( |5 - 2z| \leq 14 \) means that the expression inside the absolute value lies between \(-14\) and \(14\):

\[
-14 \leq 5 - 2z \leq 14
\]

### Step 3: Solve the compound inequality.
To solve for \(z\), first subtract 5 from all parts of the inequality:

\[
-14 - 5 \leq -2z \leq 14 - 5
\]

\[
-19 \leq -2z \leq 9
\]

Now, divide all parts of the inequality by \(-2\) and reverse the inequality signs:

\[
\frac{-19}{-2} \geq z \geq \frac{9}{-2}
\]

\[
9.5 \geq z \geq -4.5
\]

This can also be written as:

\[
-4.5 \leq z \leq 9.5
\]

### Step 4: Express the solution in interval notation.
The solution to the inequality is all values of \( z \) between \(-4.5\) and \(9.5\), inclusive. In interval notation, this is:

\[
[-4.5, 9.5]
\]

So, the solution in interval notation is \( [-4.5, 9.5] \).

Would you like to review the details or have any questions? Here are five related questions to further your understanding:

1. How does the solution change if the inequality is \( 2 - |5 - 2z| > -12 \)?
2. What if the inequality was \( 2 - |5 - 2z| \leq -12 \)? What would happen in this case?
3. How would you solve the inequality \( |3z - 7| \leq 10 \) in interval notation?
4. What are the key steps to remember when solving an absolute value inequality?
5. Can you solve \( |4z + 1| \geq 6 \) and express the solution in interval notation?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Learn how to solve the inequality 2 - |5 - 2z| ≥ -12 step-by-step with a detailed explanation and interval notation. This problem involves inequalities, absolute values, and is suitable for high school students.

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