Let's solve the equations step by step.
1) Solve for \( x \) in the equation:
\[ -2x - 9 = 15 \]
Step 1: Add 9 to both sides: \[ -2x = 15 + 9 \] \[ -2x = 24 \]
Step 2: Divide by -2: \[ x = \frac{24}{-2} \] \[ x = -12 \]
2) Solve for \( x \) in the equation:
\[ 16 - \frac{3}{4}x = 10 \]
Step 1: Subtract 16 from both sides: \[ -\frac{3}{4}x = 10 - 16 \] \[ -\frac{3}{4}x = -6 \]
Step 2: Multiply each side by -\(\frac{4}{3}\): \[ x = -6 \cdot -\frac{4}{3} \] \[ x = 8 \]
3) Solve for \( x \) in the equation:
\[ 5(2x + 4) - 4 = 2x + 8 \]
Step 1: Distribute the 5: \[ 10x + 20 - 4 = 2x + 8 \] \[ 10x + 16 = 2x + 8 \]
Step 2: Subtract \( 2x \) from both sides: \[ 10x - 2x + 16 = 8 \] \[ 8x + 16 = 8 \]
Step 3: Subtract 16 from both sides: \[ 8x = 8 - 16 \] \[ 8x = -8 \]
Step 4: Divide by 8: \[ x = -1 \]
4) Solve for \( x \) in the equation:
\[ 6x - 8 + 2x = 32 \]
Step 1: Combine like terms: \[ (6x + 2x) - 8 = 32 \] \[ 8x - 8 = 32 \]
Step 2: Add 8 to both sides: \[ 8x = 32 + 8 \] \[ 8x = 40 \]
Step 3: Divide by 8: \[ x = \frac{40}{8} \] \[ x = 5 \]
5) Write equations for comparing listens:
- Let \( x \) represent the number of days since the beginning of the month.
- The first song's listens can be represented by the equation:
\( 500 + 50x \) - The second song's listens can be represented by the equation:
\( 660 + 42x \)
We want to find \( x \) when both songs have the same number of listens: \[ 500 + 50x = 660 + 42x \]
Step 1: Subtract 42x from both sides: \[ 500 + 8x = 660 \]
Step 2: Subtract 500 from both sides: \[ 8x = 660 - 500 \] \[ 8x = 160 \]
Step 3: Divide by 8: \[ x = \frac{160}{8} \] \[ x = 20 \]
Summary of Results:
- \( x = -12 \)
- \( x = 8 \)
- \( x = -1 \)
- \( x = 5 \)
- The two songs will have the same number of listens in 20 days.