When are the following inequalities satisfied?
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When are the following inequalities satisfied?
To solve the given inequalities, we will handle each one individually and then find the valid range of where both are satisfied:
Now we combine the solutions from both inequalities:
Therefore, combining these results, the solution is the intersection of the two ranges:
, which can be expressed as .
The final solution is .
Solve the inequality:
\( 5-3x>-10 \)
Intersection (AND) means x must satisfy both inequalities at the same time. Union (OR) means x satisfies at least one inequality. For compound inequalities with 'and', always use intersection!
Write it as a compound inequality: . This shows that x is between -2 and , not including the endpoints.
You get no solution when the two conditions contradict each other, like x < 2 AND x > 5. There's no number that can be both less than 2 and greater than 5 at the same time!
No, solve each inequality separately first. Only after getting both individual solutions should you combine them to find where they overlap.
The final answer uses whichever symbol applies. If you have x ≤ 3 and x > -1, your compound inequality becomes .
Pick a test value from your solution interval and substitute it into both original inequalities. If both are true, your answer is correct!
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