To solve the compound inequality \(-2 < 5x + 3 < 13\), you can break it into two separate inequalities:
- \(-2 < 5x + 3\)
- \(5x + 3 < 13\)
Step 1: Solve the first inequality
- \(-2 < 5x + 3\)
Subtract 3 from both sides:
\[ -2 - 3 < 5x \]
\[ -5 < 5x \]
Now, divide both sides by 5:
\[ -1 < x \]
So, this simplifies to:
\[ x > -1 \]
Step 2: Solve the second inequality
- \(5x + 3 < 13\)
Subtract 3 from both sides:
\[ 5x < 13 - 3 \]
\[ 5x < 10 \]
Now, divide both sides by 5:
\[ x < 2 \]
Step 3: Combine the results
Now we have two inequalities:
- \(x > -1\)
- \(x < 2\)
Putting this together, we can express the solution as a compound inequality:
\[ -1 < x < 2 \]
Final Answer
The solution for \(x\) in the form of a compound inequality is:
\[ (-1, 2) \]
If you want to express this in terms of integers, the integers that satisfy this inequality are:
\[ 0, 1 \]
So, the integer solution can also be written as:
\[ x \in {0, 1} \]
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