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Sagot :
Let's solve the inequality [tex]\(4x + 1 \geq -9\)[/tex] and find all the integer values of [tex]\(x\)[/tex] in the interval [tex]\(-4 \leq x \leq 3\)[/tex] that satisfy it.
### Step-by-Step Solution
1. Given Inequality:
[tex]\[ 4x + 1 \geq -9 \][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we first need to get rid of the constant term on the left side. We do this by subtracting 1 from both sides:
[tex]\[ 4x + 1 - 1 \geq -9 - 1 \][/tex]
Simplifying, we get:
[tex]\[ 4x \geq -10 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Next, we divide both sides of the inequality by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{4x}{4} \geq \frac{-10}{4} \][/tex]
Simplifying, we have:
[tex]\[ x \geq -2.5 \][/tex]
4. Determine the interval for [tex]\(x\)[/tex]:
We know [tex]\(x \geq -2.5\)[/tex]. We are also given that [tex]\(x\)[/tex] must be in the interval [tex]\(-4 \leq x \leq 3\)[/tex].
5. Find the integer solutions:
Combining these two conditions, [tex]\(x\)[/tex] must satisfy both:
[tex]\[ -4 \leq x \leq 3 \][/tex]
and:
[tex]\[ x \geq -2.5 \][/tex]
To find the integer solutions, [tex]\(x\)[/tex] must be an integer within the range [tex]\([-4, 3]\)[/tex] and be greater than or equal to [tex]\(-2.5\)[/tex].
6. List all possible integer values:
The integers in the interval [tex]\([-4, 3]\)[/tex] are [tex]\(-4, -3, -2, -1, 0, 1, 2, 3\)[/tex]. From these, we exclude the values that do not satisfy [tex]\(x \geq -2.5\)[/tex]:
- [tex]\(-4\)[/tex] and [tex]\(-3\)[/tex] are less than [tex]\(-2.5\)[/tex], so they do not satisfy the inequality.
Remaining integers are: [tex]\(-2, -1, 0, 1, 2, and 3\)[/tex].
### Conclusion
The integer values of [tex]\(x\)[/tex] in the interval [tex]\(-4 \leq x \leq 3\)[/tex] that satisfy the inequality [tex]\(4x + 1 \geq -9\)[/tex] are:
[tex]\[ \boxed{-2, -1, 0, 1, 2, 3} \][/tex]
### Step-by-Step Solution
1. Given Inequality:
[tex]\[ 4x + 1 \geq -9 \][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we first need to get rid of the constant term on the left side. We do this by subtracting 1 from both sides:
[tex]\[ 4x + 1 - 1 \geq -9 - 1 \][/tex]
Simplifying, we get:
[tex]\[ 4x \geq -10 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Next, we divide both sides of the inequality by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{4x}{4} \geq \frac{-10}{4} \][/tex]
Simplifying, we have:
[tex]\[ x \geq -2.5 \][/tex]
4. Determine the interval for [tex]\(x\)[/tex]:
We know [tex]\(x \geq -2.5\)[/tex]. We are also given that [tex]\(x\)[/tex] must be in the interval [tex]\(-4 \leq x \leq 3\)[/tex].
5. Find the integer solutions:
Combining these two conditions, [tex]\(x\)[/tex] must satisfy both:
[tex]\[ -4 \leq x \leq 3 \][/tex]
and:
[tex]\[ x \geq -2.5 \][/tex]
To find the integer solutions, [tex]\(x\)[/tex] must be an integer within the range [tex]\([-4, 3]\)[/tex] and be greater than or equal to [tex]\(-2.5\)[/tex].
6. List all possible integer values:
The integers in the interval [tex]\([-4, 3]\)[/tex] are [tex]\(-4, -3, -2, -1, 0, 1, 2, 3\)[/tex]. From these, we exclude the values that do not satisfy [tex]\(x \geq -2.5\)[/tex]:
- [tex]\(-4\)[/tex] and [tex]\(-3\)[/tex] are less than [tex]\(-2.5\)[/tex], so they do not satisfy the inequality.
Remaining integers are: [tex]\(-2, -1, 0, 1, 2, and 3\)[/tex].
### Conclusion
The integer values of [tex]\(x\)[/tex] in the interval [tex]\(-4 \leq x \leq 3\)[/tex] that satisfy the inequality [tex]\(4x + 1 \geq -9\)[/tex] are:
[tex]\[ \boxed{-2, -1, 0, 1, 2, 3} \][/tex]
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