Solve for all values of x*x* in simplest form.

minus, 4, vertical bar, 2, x, minus, 5, vertical bar, plus, 5, equals, minus, 19

−4∣2*x*−5∣+5=

−19

We are given the equation:

\[
-4 |2x - 5| + 5 = -19
\]

### Step 1: Isolate the absolute value term

Subtract 5 from both sides of the equation to isolate the term involving the absolute value:

\[
-4 |2x - 5| = -19 - 5
\]
\[
-4 |2x - 5| = -24
\]

### Step 2: Divide both sides by -4

Now divide both sides of the equation by -4 to simplify:

\[
|2x - 5| = \frac{-24}{-4}
\]
\[
|2x - 5| = 6
\]

### Step 3: Solve the two cases for the absolute value

Since \( |2x - 5| = 6 \), we have two cases to solve:

#### Case 1: \( 2x - 5 = 6 \)
\[
2x - 5 = 6
\]
Add 5 to both sides:
\[
2x = 11
\]
Divide by 2:
\[
x = \frac{11}{2}
\]

#### Case 2: \( 2x - 5 = -6 \)
\[
2x - 5 = -6
\]
Add 5 to both sides:
\[
2x = -1
\]
Divide by 2:
\[
x = -\frac{1}{2}
\]

### Step 4: Final answer

The solutions are:
\[
x = \frac{11}{2} \quad \text{or} \quad x = -\frac{1}{2}
\]

Let me know if you'd like further details or have any questions!

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Here are 5 related questions:
1. How do you solve equations with absolute values in general?
2. What happens if there is a different constant outside the absolute value?
3. How can you check the solutions in the original equation?
4. What is the geometric interpretation of absolute value equations?
5. How do we deal with inequalities involving absolute values?

**Tip**: Always check your solutions by substituting them back into the original equation!

Solve for all values of x in simplest form. −4|2x−5|+5=−19

Learn how to solve the absolute value equation −4|2x−5|+5=−19 with a step-by-step solution. This problem covers key algebraic concepts, including absolute value properties and linear equation solving. Suitable for students in Grades 9-11.

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