Question: Solve absolute value equations 9LF
How many solutions does the equation have?
-5 = |4w| - 5
no solution
one solution
two solutions
1. **State the problem:** We need to find how many solutions the equation $$-5 = |4w| - 5$$ has.
2. **Isolate the absolute value expression:** Add 5 to both sides:
$$-5 + 5 = |4w| - 5 + 5$$
$$0 = |4w|$$
3. **Interpret the absolute value:** The absolute value of any expression is always non-negative, so $$|4w| = 0$$ means the expression inside the absolute value must be zero:
$$4w = 0$$
4. **Solve for $$w$$:**
$$w = \frac{0}{4} = 0$$
5. **Check the solution:** Substitute $$w=0$$ back into the original equation:
$$|4 \times 0| - 5 = 0 - 5 = -5$$
which matches the left side.
6. **Conclusion:** There is exactly one solution to the equation.
**Final answer:** one solution