Solve the following inequality for $x$.
$-26+13 x+2>2-13 x$
A. $x<2$
B. $x \geq-1$
C. $x>1$
D. $x<-2$
Answer (2)
Combine like terms: 2 - 13x"> − 26 + 13 x + 2 > 2 − 13 x becomes 2 - 13x"> − 24 + 13 x > 2 − 13 x .
Add 13 x to both sides: 2"> − 24 + 26 x > 2 .
Add 24 to both sides: 26"> 26 x > 26 .
Divide by 26 : 1"> x > 1 . The final answer is 1}"> x > 1 .
Explanation
Understanding the Inequality We are given the inequality 2 - 13x"> − 26 + 13 x + 2 > 2 − 13 x . Our goal is to solve for x , which means isolating x on one side of the inequality.
Simplifying the Left Side First, we simplify the left side of the inequality by combining the constant terms: − 26 + 2 = − 24 . So the inequality becomes 2 - 13x"> − 24 + 13 x > 2 − 13 x .
Adding 13x to Both Sides Next, we want to get all the terms with x on one side. We can add 13 x to both sides of the inequality: 2 - 13x + 13x"> − 24 + 13 x + 13 x > 2 − 13 x + 13 x 2"> − 24 + 26 x > 2
Adding 24 to Both Sides Now, we want to isolate the term with x . We can add 24 to both sides of the inequality: 2 + 24"> − 24 + 26 x + 24 > 2 + 24 26"> 26 x > 26
Dividing by 26 Finally, we divide both sides by 26 to solve for x :
\frac{{26}}{{26}}"> 26 26 x > 26 26 1"> x > 1
Final Answer The solution to the inequality is 1"> x > 1 . Comparing this to the given options, we see that option C is the correct answer.
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, suppose you are managing a budget and need to ensure that your expenses do not exceed your income. If your income is represented by I and your expenses by E , the inequality E"> I > E ensures that you stay within your budget. Solving inequalities helps you determine the limits of spending while maintaining financial stability. Similarly, in engineering, inequalities are used to set safety margins and ensure that structures can withstand certain loads or stresses.
The solution to the inequality 2 - 13x"> − 26 + 13 x + 2 > 2 − 13 x simplifies to 1"> x > 1 . Thus, the correct answer is option C: 1"> x > 1 .
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