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Chapter 2: Problem 53

Solve using the addition principle. $$ -9 x+17>17-8 x $$

Short Answer

Expert verified
x < 0

Step by step solution

01

Rewrite the Inequality

Start with the given inequality -9x + 17 > 17 - 8x
02

Isolate Variable Terms

Add 8x to both sides to move all x terms to one side: -9x + 8x + 17 > 17 - 8x + 8x Simplify to get: -x + 17 > 17
03

Simplify the Inequality

Subtract 17 from both sides to isolate the x term: -x + 17 - 17 > 17 - 17 Simplify to get: -x > 0
04

Solve for x

Divide both sides by -1, remembering to reverse the inequality sign when dividing by a negative number: -x / -1 < 0 / -1 Simplify to get: x < 0

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition Principle
The addition principle is a key concept in solving inequalities. It states that you can add or subtract the same number on both sides of an inequality without changing the inequality's truth. Start by identifying which terms you need to move:
In the inequality \(-9x + 17 > 17 - 8x\), our goal is to get all the \(x\) terms on one side.
We do this by adding \(8x\) to both sides:
\(-9x + 8x + 17 > 17 - 8x + 8x\)
Now, you can combine like terms. This method helps in maintaining balance and keeps the inequality equivalent.
Isolating the Variable
Isolating the variable is the next critical step. Here, we simplify our inequality:
\(-x + 17 > 17\)
We want to isolate \(x\), which means getting rid of the 17 on the left-hand side. To do this, we subtract 17 from both sides:
\(-x + 17 - 17 > 17 - 17\)
This simplifies to:
\(-x > 0\)
You can see that our variable term \(-x\) is now alone on one side of the inequality.
Division by a Negative Number
When solving inequalities, dividing by a negative number requires special attention. In our case, we need to get rid of the negative sign in front of \(x\):
\(-x > 0\)
We do this by dividing both sides by \(-1\):
\(-x / -1 < 0 / -1\)
This leaves us with:
\(-x / -1 = x \) and \((0)/(-1) = 0\)
So we have:
\(x < 0\)
Dividing by a negative number is unique because it involves reversing the inequality sign.
Reversing the Inequality Sign
Reversing the inequality sign is crucial when dividing or multiplying both sides of an inequality by a negative number. For instance, in our step where we divided \(-x > 0\) by \(-1\), we had to switch the > sign to a < sign. Hence, our final inequality after division becomes:
\(x < 0\)
Always remember that inequality signs reverse only when you multiply or divide both sides of an inequality by a negative number. This ensures the inequality remains true.

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