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Sagot :
### Step-by-Step Solution
#### Part 1: Solving the Inequality [tex]\(5x - 2 \leq 12x - 16\)[/tex]
Given the set of numbers: [tex]\( \{3, 3, 5, 6\} \)[/tex]
1. Start with the inequality:
[tex]\[ 5x - 2 \leq 12x - 16 \][/tex]
2. Rearrange the inequality to isolate [tex]\(x\)[/tex] on one side:
[tex]\[ 5x - 2 \leq 12x - 16 \][/tex]
[tex]\[ 5x - 2 - 12x \leq -16 \][/tex]
[tex]\[ -7x - 2 \leq -16 \][/tex]
[tex]\[ -7x \leq -14 \][/tex]
3. Divide by [tex]\(-7\)[/tex] (remember, dividing by a negative number reverses the inequality):
[tex]\[ x \geq 2 \][/tex]
Now check which values in the given set satisfy the inequality [tex]\( x \geq 2 \)[/tex]:
- [tex]\(3\)[/tex] (meets the condition)
- [tex]\(3\)[/tex] (meets the condition)
- [tex]\(5\)[/tex] (meets the condition)
- [tex]\(6\)[/tex] (meets the condition)
Thus, the set of values satisfying the inequality is: [tex]\(\{3, 5, 6\}\)[/tex]
#### Part 2: Evaluating [tex]\(\frac{x}{2x + 1}\)[/tex] When [tex]\(x = 2\)[/tex]
1. Substitute [tex]\(x = 2\)[/tex] into the expression:
[tex]\[ \frac{2}{2(2) + 1} = \frac{2}{4 + 1} = \frac{2}{5} \][/tex]
2. Calculate the fraction:
[tex]\[ \frac{2}{5} = 0.4 \][/tex]
So, [tex]\(\frac{x}{2x + 1}\)[/tex] when [tex]\(x = 2\)[/tex] is [tex]\(0.4\)[/tex].
#### Part 3: Simplifying the Expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex]
1. First, rewrite the given expression:
[tex]\[ -\frac{2x - 3y}{3} - \frac{x + y}{2} \][/tex]
2. Combine the terms by finding a common denominator:
[tex]\[ -\frac{2x - 3y}{3} = -\frac{2x}{3} + \frac{3y}{3} = -\frac{2x}{3} + y \][/tex]
[tex]\[ -\frac{x + y}{2} = -\frac{x}{2} - \frac{y}{2} \][/tex]
3. Combine the two parts, using a common denominator of 6:
[tex]\[ -\frac{2x}{3} - \frac{x}{2} = -\frac{2x \cdot 2}{6} - \frac{x \cdot 3}{6} = -\frac{4x}{6} - \frac{3x}{6} = -\frac{7x}{6} \][/tex]
[tex]\[ \frac{3y}{3} - \frac{y}{2} = y - \frac{y \cdot 3}{6} = y - \frac{3y}{6} = y - \frac{y}{2} = \frac{2y}{2} - \frac{3y}{6} = -\frac{3y}{6} = -\frac{3y}{2} \][/tex]
4. Combine these final terms together:
[tex]\[ -\frac{7x}{6} - \frac{3y}{2} \][/tex]
Thus, the simplified form of the expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex] is [tex]\(-\frac{7x}{6} - \frac{3y}{2}\)[/tex].
### Final Answers
1. The values of [tex]\(x\)[/tex] that satisfy the inequality [tex]\(5x - 2 \leq 12x - 16\)[/tex] are: [tex]\(\{3, 5, 6\}\)[/tex]
2. The value of [tex]\(\frac{x}{2x + 1}\)[/tex] when [tex]\(x = 2\)[/tex] is 0.4.
3. The simplified form of the expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex] is [tex]\(-\frac{7x}{6} - \frac{3y}{2}\)[/tex].
#### Part 1: Solving the Inequality [tex]\(5x - 2 \leq 12x - 16\)[/tex]
Given the set of numbers: [tex]\( \{3, 3, 5, 6\} \)[/tex]
1. Start with the inequality:
[tex]\[ 5x - 2 \leq 12x - 16 \][/tex]
2. Rearrange the inequality to isolate [tex]\(x\)[/tex] on one side:
[tex]\[ 5x - 2 \leq 12x - 16 \][/tex]
[tex]\[ 5x - 2 - 12x \leq -16 \][/tex]
[tex]\[ -7x - 2 \leq -16 \][/tex]
[tex]\[ -7x \leq -14 \][/tex]
3. Divide by [tex]\(-7\)[/tex] (remember, dividing by a negative number reverses the inequality):
[tex]\[ x \geq 2 \][/tex]
Now check which values in the given set satisfy the inequality [tex]\( x \geq 2 \)[/tex]:
- [tex]\(3\)[/tex] (meets the condition)
- [tex]\(3\)[/tex] (meets the condition)
- [tex]\(5\)[/tex] (meets the condition)
- [tex]\(6\)[/tex] (meets the condition)
Thus, the set of values satisfying the inequality is: [tex]\(\{3, 5, 6\}\)[/tex]
#### Part 2: Evaluating [tex]\(\frac{x}{2x + 1}\)[/tex] When [tex]\(x = 2\)[/tex]
1. Substitute [tex]\(x = 2\)[/tex] into the expression:
[tex]\[ \frac{2}{2(2) + 1} = \frac{2}{4 + 1} = \frac{2}{5} \][/tex]
2. Calculate the fraction:
[tex]\[ \frac{2}{5} = 0.4 \][/tex]
So, [tex]\(\frac{x}{2x + 1}\)[/tex] when [tex]\(x = 2\)[/tex] is [tex]\(0.4\)[/tex].
#### Part 3: Simplifying the Expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex]
1. First, rewrite the given expression:
[tex]\[ -\frac{2x - 3y}{3} - \frac{x + y}{2} \][/tex]
2. Combine the terms by finding a common denominator:
[tex]\[ -\frac{2x - 3y}{3} = -\frac{2x}{3} + \frac{3y}{3} = -\frac{2x}{3} + y \][/tex]
[tex]\[ -\frac{x + y}{2} = -\frac{x}{2} - \frac{y}{2} \][/tex]
3. Combine the two parts, using a common denominator of 6:
[tex]\[ -\frac{2x}{3} - \frac{x}{2} = -\frac{2x \cdot 2}{6} - \frac{x \cdot 3}{6} = -\frac{4x}{6} - \frac{3x}{6} = -\frac{7x}{6} \][/tex]
[tex]\[ \frac{3y}{3} - \frac{y}{2} = y - \frac{y \cdot 3}{6} = y - \frac{3y}{6} = y - \frac{y}{2} = \frac{2y}{2} - \frac{3y}{6} = -\frac{3y}{6} = -\frac{3y}{2} \][/tex]
4. Combine these final terms together:
[tex]\[ -\frac{7x}{6} - \frac{3y}{2} \][/tex]
Thus, the simplified form of the expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex] is [tex]\(-\frac{7x}{6} - \frac{3y}{2}\)[/tex].
### Final Answers
1. The values of [tex]\(x\)[/tex] that satisfy the inequality [tex]\(5x - 2 \leq 12x - 16\)[/tex] are: [tex]\(\{3, 5, 6\}\)[/tex]
2. The value of [tex]\(\frac{x}{2x + 1}\)[/tex] when [tex]\(x = 2\)[/tex] is 0.4.
3. The simplified form of the expression [tex]\(-\frac{2x - 3y}{3} - \frac{x + y}{2}\)[/tex] is [tex]\(-\frac{7x}{6} - \frac{3y}{2}\)[/tex].
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