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Start your 14-day free trial to unlock the full solution →The solution set is the intersection of two linear inequalities: from the first and from the second. The stricter condition dominates, so the final answer is .
Concept and Intuition
When you have two inequalities joined by an implied "and" (as in a system), you're looking for values of that satisfy both conditions simultaneously. Think of it like two gates: you can only pass through if both gates are open. The solution is the overlap — the intersection — of the two individual solution sets.
Each inequality is linear, meaning it describes a half-line on the number line. Solving them separately is straightforward: isolate by performing the same operations on both sides, being careful with the direction of the inequality when multiplying or dividing by a negative number. Then, find where the two half-lines overlap.
A common mistake is to solve each inequality correctly but then take the union instead of the intersection. Remember: "and" means both must hold — you need the region common to both.
Step-by-Step Solution
1. Solve the first inequality:
Start by expanding the right-hand side:
Now bring the term to the left and the to the right (or vice versa — just keep it tidy):
So the first condition is . On the number line, this is all numbers to the right of , not including itself.
2. Solve the second inequality:
Add to both sides to get the terms together:
Now subtract from both sides:
So the second condition is . This is all numbers to the right of .
3. Find the intersection of the two solution sets
We need that satisfies both and .
Visualise the number line: …
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