Let's analyze and solve each equation step by step, determining their solutions and arranging them in increasing order based on these solutions.
1. First equation:
[tex]\[
\frac{1}{4} x + \frac{5}{2} x - 2 = 4 - \frac{1}{4} x
\][/tex]
Combining like terms:
[tex]\[
\left(\frac{1}{4} + \frac{5}{2} + \frac{1}{4}\right)x - 2 = 4
\][/tex]
Simplifies to:
[tex]\[
3x - 2 = 4
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
3x = 6 \implies x = 2
\][/tex]
2. Second equation:
[tex]\[
7.9x + x + 4 = -1.1x - 16
\][/tex]
Combining like terms:
[tex]\[
8.9x + 4 = -1.1x - 16
\][/tex]
Combining the [tex]\( x \)[/tex] terms and constants:
[tex]\[
10x = -20 \implies x = -2
\][/tex]
3. Third equation:
[tex]\[
3.2x + 5.7 = -2.5x
\][/tex]
Combining like terms:
[tex]\[
3.2x + 2.5x = -5.7 \implies 5.7x = -5.7 \implies x = -1
\][/tex]
4. Fourth equation:
[tex]\[
10.1x - 1.6x + 44 = -7
\][/tex]
Simplifying the terms:
[tex]\[
8.5x + 44 = -7 \implies 8.5x = -51 \implies x = -6
\][/tex]
Now, let's arrange these solutions in increasing order:
- Equation 4: [tex]\( x = -6 \)[/tex]
- Equation 2: [tex]\( x = -2 \)[/tex]
- Equation 3: [tex]\( x = -1 \)[/tex]
- Equation 1: [tex]\( x = 2 \)[/tex]
Thus, the equations in increasing order of the value of their solutions are:
1. [tex]\( 10.1x - 1.6x + 44 = -7 \)[/tex] (Solution: [tex]\( x = -6 \)[/tex])
2. [tex]\( 7.9x + x + 4 = -1.1x - 16 \)[/tex] (Solution: [tex]\( x = -2 \)[/tex])
3. [tex]\( 3.2x + 5.7 = -2.5x \)[/tex] (Solution: [tex]\( x = -1 \)[/tex])
4. [tex]\( \frac{1}{4}x + \frac{5}{2}x - 2 = 4 - \frac{1}{4}x \)[/tex] (Solution: [tex]\( x = 2 \)[/tex])
