Solve: $-5<-7-2 x<4$
The answer is $\square$ $
Answer (1)
Add 7 to all parts of the inequality: 2 < − 2 x < 11 .
Divide all parts by -2, remembering to flip the inequality signs: x > -\frac{11}{2}"> − 1 > x > − 2 11 .
Rewrite the inequality in standard form: − 2 11 < x < − 1 .
The solution is: − 2 11 < x < − 1 .
Explanation
Understanding the Inequality We are given the inequality − 5 < − 7 − 2 x < 4 . Our goal is to isolate x in the middle to solve for its possible values.
Adding 7 to All Parts First, we add 7 to all three parts of the inequality to simplify it: − 5 + 7 < − 7 − 2 x + 7 < 4 + 7 This simplifies to: 2 < − 2 x < 11
Dividing by -2 and Reversing Signs Next, we divide all three parts of the inequality by -2. Remember that when we divide by a negative number, we must reverse the direction of the inequality signs: \frac{-2x}{-2} > \frac{11}{-2}"> − 2 2 > − 2 − 2 x > − 2 11 This simplifies to: x > -\frac{11}{2}"> − 1 > x > − 2 11
Rewriting the Inequality Finally, we rewrite the inequality so that the smaller number is on the left and the larger number is on the right. This gives us: − 2 11 < x < − 1
Examples
Imagine you're trying to keep the temperature of a chemical reaction within a specific range for optimal results. This problem is similar to finding the range of a variable (like temperature) that satisfies certain conditions. Understanding how to solve inequalities helps you determine the acceptable range for the variable to ensure the reaction proceeds as desired. This is crucial in many scientific and engineering applications where maintaining specific conditions is essential for success.
