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Sagot :
To determine which compound inequality matches the given number line, we need to solve each inequality algebraically and analyze the solution sets.
### Inequality 1: [tex]\( -4 x \geq 12 \)[/tex] and [tex]\( -4 x < -8 \)[/tex]
Solving [tex]\( -4 x \geq 12 \)[/tex]:
[tex]\[ -4 x \geq 12 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Solving [tex]\( -4 x < -8 \)[/tex]:
[tex]\[ -4 x < -8 \][/tex]
[tex]\[ x > 2 \][/tex]
Combining the two solutions:
[tex]\[ x \leq -3 \quad \text{and} \quad x > 2 \][/tex]
Since there is no overlap between [tex]\( x \leq -3 \)[/tex] and [tex]\( x > 2 \)[/tex], there is no solution for this compound inequality.
### Inequality 2: [tex]\( 2 x \leq -6 \)[/tex] or [tex]\( 2 x \geq 4 \)[/tex]
Solving [tex]\( 2 x \leq -6 \)[/tex]:
[tex]\[ 2 x \leq -6 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Solving [tex]\( 2 x \geq 4 \)[/tex]:
[tex]\[ 2 x \geq 4 \][/tex]
[tex]\[ x \geq 2 \][/tex]
Combining the two solutions:
[tex]\[ x \leq -3 \quad \text{or} \quad x \geq 2 \][/tex]
This means [tex]\( x \in (-\infty, -3] \cup [2, \infty) \)[/tex] which represents two disjoint intervals.
### Inequality 3: [tex]\( -13 \leq 4 x - 1 < 7 \)[/tex]
Breaking it into two parts and solving:
Part 1: [tex]\( -13 \leq 4 x - 1 \)[/tex]
[tex]\[ -13 \leq 4 x - 1 \][/tex]
[tex]\[ -12 \leq 4 x \][/tex]
[tex]\[ -3 \leq x \][/tex]
Part 2: [tex]\( 4 x - 1 < 7 \)[/tex]
[tex]\[ 4 x - 1 < 7 \][/tex]
[tex]\[ 4 x < 8 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two solutions:
[tex]\[ -3 \leq x < 2 \][/tex]
This represents a continuous interval from [tex]\(-3\)[/tex] to [tex]\(2\)[/tex].
### Inequality 4: [tex]\( 5 x \geq -15 \)[/tex] or [tex]\( 5 x < 10 \)[/tex]
Solving [tex]\( 5 x \geq -15 \)[/tex]:
[tex]\[ 5 x \geq -15 \][/tex]
[tex]\[ x \geq -3 \][/tex]
Solving [tex]\( 5 x < 10 \)[/tex]:
[tex]\[ 5 x < 10 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two solutions:
[tex]\[ x \geq -3 \quad \text{or} \quad x < 2 \][/tex]
This inequality is not a standard compound inequality as it states [tex]\( x \)[/tex] can be any value, which is always true – essentially it doesn't restrict [tex]\(x\)[/tex].
### Conclusion
The number line that is likely represented by one of these inequalities is:
[tex]\[ -3 \leq x < 2 \][/tex]
This corresponds to Inequality 3: [tex]\( -13 \leq 4 x - 1 < 7 \)[/tex].
Thus, the correct answer is:
[tex]\[ -13 \leq 4 x - 1 < 7 \][/tex]
### Inequality 1: [tex]\( -4 x \geq 12 \)[/tex] and [tex]\( -4 x < -8 \)[/tex]
Solving [tex]\( -4 x \geq 12 \)[/tex]:
[tex]\[ -4 x \geq 12 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Solving [tex]\( -4 x < -8 \)[/tex]:
[tex]\[ -4 x < -8 \][/tex]
[tex]\[ x > 2 \][/tex]
Combining the two solutions:
[tex]\[ x \leq -3 \quad \text{and} \quad x > 2 \][/tex]
Since there is no overlap between [tex]\( x \leq -3 \)[/tex] and [tex]\( x > 2 \)[/tex], there is no solution for this compound inequality.
### Inequality 2: [tex]\( 2 x \leq -6 \)[/tex] or [tex]\( 2 x \geq 4 \)[/tex]
Solving [tex]\( 2 x \leq -6 \)[/tex]:
[tex]\[ 2 x \leq -6 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Solving [tex]\( 2 x \geq 4 \)[/tex]:
[tex]\[ 2 x \geq 4 \][/tex]
[tex]\[ x \geq 2 \][/tex]
Combining the two solutions:
[tex]\[ x \leq -3 \quad \text{or} \quad x \geq 2 \][/tex]
This means [tex]\( x \in (-\infty, -3] \cup [2, \infty) \)[/tex] which represents two disjoint intervals.
### Inequality 3: [tex]\( -13 \leq 4 x - 1 < 7 \)[/tex]
Breaking it into two parts and solving:
Part 1: [tex]\( -13 \leq 4 x - 1 \)[/tex]
[tex]\[ -13 \leq 4 x - 1 \][/tex]
[tex]\[ -12 \leq 4 x \][/tex]
[tex]\[ -3 \leq x \][/tex]
Part 2: [tex]\( 4 x - 1 < 7 \)[/tex]
[tex]\[ 4 x - 1 < 7 \][/tex]
[tex]\[ 4 x < 8 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two solutions:
[tex]\[ -3 \leq x < 2 \][/tex]
This represents a continuous interval from [tex]\(-3\)[/tex] to [tex]\(2\)[/tex].
### Inequality 4: [tex]\( 5 x \geq -15 \)[/tex] or [tex]\( 5 x < 10 \)[/tex]
Solving [tex]\( 5 x \geq -15 \)[/tex]:
[tex]\[ 5 x \geq -15 \][/tex]
[tex]\[ x \geq -3 \][/tex]
Solving [tex]\( 5 x < 10 \)[/tex]:
[tex]\[ 5 x < 10 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two solutions:
[tex]\[ x \geq -3 \quad \text{or} \quad x < 2 \][/tex]
This inequality is not a standard compound inequality as it states [tex]\( x \)[/tex] can be any value, which is always true – essentially it doesn't restrict [tex]\(x\)[/tex].
### Conclusion
The number line that is likely represented by one of these inequalities is:
[tex]\[ -3 \leq x < 2 \][/tex]
This corresponds to Inequality 3: [tex]\( -13 \leq 4 x - 1 < 7 \)[/tex].
Thus, the correct answer is:
[tex]\[ -13 \leq 4 x - 1 < 7 \][/tex]
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