Answer :
### First System of Inequalities
Let's tackle each set of inequalities step-by-step.
Step 1: Solve the system [tex]\(-2x + 3 < 5\)[/tex] and [tex]\(-4x - 3 > 9\)[/tex]
1. Solve [tex]\(-2x + 3 < 5\)[/tex]:
[tex]\[ \begin{align*} -2x + 3 &< 5 \\ -2x &< 2 \\ x &> -1 \end{align*} \][/tex]
2. Solve [tex]\(-4x - 3 > 9\)[/tex]:
[tex]\[ \begin{align*} -4x - 3 &> 9 \\ -4x &> 12 \\ x &< -3 \end{align*} \][/tex]
Thus, we have [tex]\( x > -1 \)[/tex] and [tex]\( x < -3 \)[/tex]. There is no [tex]\( x \)[/tex] that satisfies both conditions simultaneously.
### Second System of Inequalities
Step 2: Solve the system [tex]\(\frac{x}{2} - 2 > 0\)[/tex] and [tex]\(-4x + 2 < -6\)[/tex]
1. Solve [tex]\(\frac{x}{2} - 2 > 0\)[/tex]:
[tex]\[ \begin{align*} \frac{x}{2} - 2 &> 0 \\ \frac{x}{2} &> 2 \\ x &> 4 \end{align*} \][/tex]
2. Solve [tex]\(-4x + 2 < -6\)[/tex]:
[tex]\[ \begin{align*} -4x + 2 &< -6 \\ -4x &< -8 \\ x &> 2 \end{align*} \][/tex]
So, the solution to this system is the intersection: [tex]\( x > 4 \)[/tex].
### Third Inequality
Step 3: Solve the inequality [tex]\(2(x + 3) < -2\)[/tex] or [tex]\(-4x + 3 < -13\)[/tex]
1. Solve [tex]\(2(x + 3) < -2\)[/tex]:
[tex]\[ \begin{align*} 2(x + 3) &< -2 \\ x + 3 &< -1 \\ x &< -4 \end{align*} \][/tex]
2. Solve [tex]\(-4x + 3 < -13\)[/tex]:
[tex]\[ \begin{align*} -4x + 3 &< -13 \\ -4x &< -16 \\ x &> 4 \end{align*} \][/tex]
The solution is [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex].
### Fourth Compound Inequality
Step 4: Solve the compound inequality [tex]\(-10 < -3x + 2 < 14\)[/tex]
1. Solve [tex]\(-10 < -3x + 2\)[/tex]:
[tex]\[ \begin{align*} -10 &< -3x + 2 \\ -12 &< -3x \\ 4 &> x \\ x &< 4 \end{align*} \][/tex]
2. Solve [tex]\(-3x + 2 < 14\)[/tex]:
[tex]\[ \begin{align*} -3x + 2 &< 14 \\ -3x &< 12 \\ x &> -4 \end{align*} \][/tex]
The solution is [tex]\(-4 < x < 4\)[/tex].
### Final Solutions
Summarize solutions for each system/inquality:
1. No [tex]\(x\)[/tex] satisfies both inequalities: [tex]\( \emptyset \)[/tex]
2. [tex]\( x > 4 \)[/tex]
3. [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex]
4. [tex]\( -4 < x < 4 \)[/tex]
Therefore, the detailed solutions are:
1. No solution, [tex]\( \text{False} \)[/tex].
2. [tex]\( x > 4 \)[/tex]
3. [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex]
4. [tex]\( -4 < x < 4 \)[/tex].
These are the results for each system of inequalities given.
Let's tackle each set of inequalities step-by-step.
Step 1: Solve the system [tex]\(-2x + 3 < 5\)[/tex] and [tex]\(-4x - 3 > 9\)[/tex]
1. Solve [tex]\(-2x + 3 < 5\)[/tex]:
[tex]\[ \begin{align*} -2x + 3 &< 5 \\ -2x &< 2 \\ x &> -1 \end{align*} \][/tex]
2. Solve [tex]\(-4x - 3 > 9\)[/tex]:
[tex]\[ \begin{align*} -4x - 3 &> 9 \\ -4x &> 12 \\ x &< -3 \end{align*} \][/tex]
Thus, we have [tex]\( x > -1 \)[/tex] and [tex]\( x < -3 \)[/tex]. There is no [tex]\( x \)[/tex] that satisfies both conditions simultaneously.
### Second System of Inequalities
Step 2: Solve the system [tex]\(\frac{x}{2} - 2 > 0\)[/tex] and [tex]\(-4x + 2 < -6\)[/tex]
1. Solve [tex]\(\frac{x}{2} - 2 > 0\)[/tex]:
[tex]\[ \begin{align*} \frac{x}{2} - 2 &> 0 \\ \frac{x}{2} &> 2 \\ x &> 4 \end{align*} \][/tex]
2. Solve [tex]\(-4x + 2 < -6\)[/tex]:
[tex]\[ \begin{align*} -4x + 2 &< -6 \\ -4x &< -8 \\ x &> 2 \end{align*} \][/tex]
So, the solution to this system is the intersection: [tex]\( x > 4 \)[/tex].
### Third Inequality
Step 3: Solve the inequality [tex]\(2(x + 3) < -2\)[/tex] or [tex]\(-4x + 3 < -13\)[/tex]
1. Solve [tex]\(2(x + 3) < -2\)[/tex]:
[tex]\[ \begin{align*} 2(x + 3) &< -2 \\ x + 3 &< -1 \\ x &< -4 \end{align*} \][/tex]
2. Solve [tex]\(-4x + 3 < -13\)[/tex]:
[tex]\[ \begin{align*} -4x + 3 &< -13 \\ -4x &< -16 \\ x &> 4 \end{align*} \][/tex]
The solution is [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex].
### Fourth Compound Inequality
Step 4: Solve the compound inequality [tex]\(-10 < -3x + 2 < 14\)[/tex]
1. Solve [tex]\(-10 < -3x + 2\)[/tex]:
[tex]\[ \begin{align*} -10 &< -3x + 2 \\ -12 &< -3x \\ 4 &> x \\ x &< 4 \end{align*} \][/tex]
2. Solve [tex]\(-3x + 2 < 14\)[/tex]:
[tex]\[ \begin{align*} -3x + 2 &< 14 \\ -3x &< 12 \\ x &> -4 \end{align*} \][/tex]
The solution is [tex]\(-4 < x < 4\)[/tex].
### Final Solutions
Summarize solutions for each system/inquality:
1. No [tex]\(x\)[/tex] satisfies both inequalities: [tex]\( \emptyset \)[/tex]
2. [tex]\( x > 4 \)[/tex]
3. [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex]
4. [tex]\( -4 < x < 4 \)[/tex]
Therefore, the detailed solutions are:
1. No solution, [tex]\( \text{False} \)[/tex].
2. [tex]\( x > 4 \)[/tex]
3. [tex]\( x < -4 \)[/tex] or [tex]\( x > 4 \)[/tex]
4. [tex]\( -4 < x < 4 \)[/tex].
These are the results for each system of inequalities given.
