Sure! Let's go through each problem step by step to solve for the variables.
### Problem 1: [tex]\( 2 < 5\alpha + 4 = 3(2\alpha + 3 - \alpha) \)[/tex]
First, correct the compound inequality to two separate inequalities:
[tex]\[ 2 < 5\alpha + 4 \quad \text{and} \quad 5\alpha + 4 = 3(2\alpha + 3 - \alpha) \][/tex]
Solve the first inequality:
[tex]\[ 2 < 5\alpha + 4 \][/tex]
[tex]\[ 2 - 4 < 5\alpha \][/tex]
[tex]\[ -2 < 5\alpha \][/tex]
[tex]\[ \alpha > -\frac{2}{5} \][/tex]
For the equation:
[tex]\[ 5\alpha + 4 = 3(2\alpha + 3 - \alpha) \][/tex]
Simplify inside the parentheses:
[tex]\[ 5\alpha + 4 = 3(\alpha + 3) \][/tex]
Expand:
[tex]\[ 5\alpha + 4 = 3\alpha + 9 \][/tex]
Subtract [tex]\( 3\alpha \)[/tex] from both sides:
[tex]\[ 2\alpha + 4 = 9 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2\alpha = 5 \][/tex]
Divide by 2:
[tex]\[ \alpha = \frac{5}{2} \][/tex]
### Problem 2: [tex]\( 4 - 2 \cdot (2x - 3) - (2x + 4) = -2(24) \)[/tex]
Expand and simplify:
[tex]\[ 4 - 2(2x - 3) - (2x + 4) = -48 \][/tex]
Distribute the 2:
[tex]\[ 4 - 4x + 6 - 2x - 4 = -48 \][/tex]
Combine like terms:
[tex]\[ 10 - 6x = -48 \][/tex]
Subtract 10 from both sides:
[tex]\[ -6x = -58 \][/tex]
Divide by -6:
[tex]\[ x = 9 \][/tex]
### Problem 3: [tex]\( 8m - 8 - 2m + 8 = 0 \)[/tex]
Combine like terms:
[tex]\[ 6m = 0 \][/tex]
Divide by 6:
[tex]\[ m = 0 \][/tex]
### Problem 4: [tex]\( 3x + 16 = 2 - x - 6 \)[/tex]
Combine like terms on the right side:
[tex]\[ 3x + 16 = -x - 4 \][/tex]
Add [tex]\( x \)[/tex] to both sides:
[tex]\[ 4x + 16 = -4 \][/tex]
Subtract 16 from both sides:
[tex]\[ 4x = -20 \][/tex]
Divide by 4:
[tex]\[ x = -5 \][/tex]
### Problem 5: [tex]\( 3 - (a + 1) - 2 = -3x \)[/tex]
Simplify inside the parentheses:
[tex]\[ 3 - a - 1 - 2 = -3x \][/tex]
Combine like terms:
[tex]\[ 0 - a = -3x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{a}{3} \][/tex]
### Summary of Solutions
1. [tex]\( \alpha = \frac{5}{2} \)[/tex]
2. [tex]\( x = 9 \)[/tex]
3. [tex]\( m = 0 \)[/tex]
4. [tex]\( x = -5 \)[/tex]
5. [tex]\( x = \frac{a}{3} \)[/tex]
There you go! These are the solutions step-by-step.
