The image you uploaded provides questions related to absolute value equations. Let me address them one by one:

1. When does an absolute value equation have no solution?

   • Explanation: An absolute value equation has no solution when the absolute value expression is set equal to a negative number. This is because the absolute value of any real number is non-negative, meaning it cannot be less than zero.
   • Example: $|x + 2| = -5$
     • The equation $|x + 2| \geq 0$, but it is equal to $-5$, which is impossible. Therefore, no solution exists.

2. When does an absolute value equation have exactly one solution?

   • Explanation: An absolute value equation has exactly one solution when the expression inside the absolute value is equal to zero or when both sides of the equation are equal to the same positive number.
   • Example: $|x| = 0$
     • Here, $x = 0$, so the only solution is $x = 0$.

3. When does an absolute value equation have two solutions?

   • Explanation: An absolute value equation has two solutions when the absolute value is set equal to a positive number. This occurs because the expression inside the absolute value could either be positive or negative, both leading to the same absolute value.
   • Example: $|x - 3| = 4$
     • This results in two cases:
       1. $x - 3 = 4 \Rightarrow x = 7$
       2. $x - 3 = -4 \Rightarrow x = -1$ Therefore, the two solutions are $x = 7$ and $x = -1$.

4. How does the solving process change when there is an absolute value expression on both sides of the equation? Let's solve the equation $|2x + 3| = |x - 4|$:

   Step-by-Step Solution:

   To solve this, we break it down into two cases:

   Case 1: $2x + 3 = x - 4$

   • Subtract $x$ from both sides: $2x - x + 3 = -4$
   • Simplify: $x + 3 = -4$
   • Subtract 3 from both sides: $x = -7$

   Case 2: $2x + 3 = -(x - 4)$

   • Distribute the negative sign: $2x + 3 = -x + 4$
   • Add $x$ to both sides: $2x + x + 3 = 4$
   • Simplify: $3x + 3 = 4$
   • Subtract 3 from both sides: $3x = 1$
   • Divide by 3: $x = \frac{1}{3}$

   Checking the Solutions:

   • For $x = -7$:
     • $|2(-7) + 3| = |-14 + 3| = |-11| = 11$
     • $|-7 - 4| = |-11| = 11$, so this is a valid solution.
   • For $x = \frac{1}{3}$:
     • $|2(\frac{1}{3}) + 3| = |\frac{2}{3} + 3| = |\frac{11}{3}| = \frac{11}{3}$
     • $|\frac{1}{3} - 4| = |\frac{1}{3} - \frac{12}{3}| = |\frac{-11}{3}| = \frac{11}{3}$, so this is also a valid solution.

   Therefore, the solutions are $x = -7$ and $x = \frac{1}{3}$.

Structure and Number of Solutions:

The structure of the absolute value equation affects the number of solutions because:

• If the equation is set equal to a negative number, there is no solution.
• If it's set equal to zero, there is exactly one solution.
• If it's set equal to a positive number, there are two solutions (one positive and one negative expression).

Would you like further clarifications or any detailed explanations on these steps?

Related Questions:

1. How can you visualize absolute value equations on a graph?
2. What are the common applications of absolute value equations in real life?
3. How does the structure change when there are additional constants outside the absolute value?
4. Can there be more than two solutions for any absolute value equation?
5. How would you solve absolute value inequalities?

Tip:

Always remember that the absolute value of a number is its distance from zero, so it can never be negative. This is key to determining the number of solutions.