Solve the following inequality and leave your answer in interval notation:
$-1 \leq \frac{3-5 x}{2} \leq 9$
Answer (2)
Multiply all parts of the inequality by 2: − 2 ≤ 3 − 5 x ≤ 18 .
Subtract 3 from all parts: − 5 ≤ − 5 x ≤ 15 .
Divide by -5 (and reverse inequality signs): 1 ≥ x ≥ − 3 .
Express the solution in interval notation: [ − 3 , 1 ] .
Explanation
Understanding the Inequality We are given the compound inequality − 1 ≤ 2 3 − 5 x ≤ 9 . Our goal is to isolate x in the middle and express the solution in interval notation.
Multiplying by 2 First, we multiply all parts of the inequality by 2 to eliminate the fraction: 2 × ( − 1 ) ≤ 2 × 2 3 − 5 x ≤ 2 × 9
This simplifies to: − 2 ≤ 3 − 5 x ≤ 18
Subtracting 3 Next, we subtract 3 from all parts of the inequality to isolate the term with x : − 2 − 3 ≤ 3 − 5 x − 3 ≤ 18 − 3
This simplifies to: − 5 ≤ − 5 x ≤ 15
Dividing by -5 (and reversing inequality signs) Now, we divide all parts of the inequality by -5. Remember that when we divide by a negative number, we must reverse the inequality signs: − 5 − 5 ≥ − 5 − 5 x ≥ − 5 15
This simplifies to: 1 ≥ x ≥ − 3
Rewriting the Inequality We can rewrite this inequality so that the smaller number is on the left: − 3 ≤ x ≤ 1
Interval Notation Finally, we express the solution in interval notation. Since x is greater than or equal to -3 and less than or equal to 1, the interval includes both endpoints. Therefore, the solution in interval notation is [ − 3 , 1 ] .
Examples
Imagine you are setting up a temperature range for a sensitive chemical experiment. The inequality − 1 ≤ 2 3 − 5 x ≤ 9 could represent the acceptable temperature range, where x is a controllable parameter. Solving this inequality helps you determine the precise range of x values that keep the experiment within the safe temperature limits. This ensures the experiment runs correctly and safely, preventing any unwanted reactions or damages. Understanding and manipulating inequalities is crucial in many scientific and engineering applications to maintain controlled conditions.
The solution to the inequality − 1 ≤ 2 3 − 5 x ≤ 9 is found by isolating x through a series of steps, leading us to the interval [ − 3 , 1 ] . This indicates that x can take any value from -3 to 1, including the endpoints. The final answer in interval notation is [ − 3 , 1 ] .
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