What is the solution in interval notation to $-4 \leq 3 x-7<8$?
Answer (2)
Add 7 to all parts of the inequality: 3 ≤ 3 x < 15 .
Divide all parts of the inequality by 3: 1 ≤ x < 5 .
Express the solution in interval notation.
The solution in interval notation is [ 1 , 5 ) .
Explanation
Understanding the Problem We are given the inequality − 4 ≤ 3 x − 7 < 8 . We want to find the solution in interval notation.
Isolating x To solve the compound inequality, we need to isolate x in the middle. We can do this by performing the same operations on all parts of the inequality. First, we add 7 to all parts of the inequality: − 4 + 7 ≤ 3 x − 7 + 7 < 8 + 7 This simplifies to 3 ≤ 3 x < 15
Solving for x Next, we divide all parts of the inequality by 3: 3 3 ≤ 3 3 x < 3 15 This simplifies to 1 ≤ x < 5
Interval Notation Now we express the solution in interval notation. Since x is greater than or equal to 1 and less than 5, the interval notation is [ 1 , 5 ) .
Examples
Understanding and solving inequalities is crucial in various real-life situations. For instance, when budgeting, you might want to ensure your expenses ( x ) stay within a certain range, say between $100 and $500 ( 100 ≤ x ≤ 500 ). Similarly, in cooking, maintaining the oven temperature within a specific range is vital for baking the perfect cake. Inequalities also play a key role in fields like engineering, where certain parameters must fall within specific limits to ensure safety and efficiency.
The solution to the inequality − 4 ≤ 3 x − 7 < 8 is found by first isolating x, resulting in the expression 1 ≤ x < 5 . This translates to the interval notation [ 1 , 5 ) . Thus, the final answer is [ 1 , 5 ) .
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