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Chapter 2: Problem 30

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ 20>-2 u+40 $$

Short Answer

Expert verified
u > 10

Step by step solution

01

- Isolate the variable term

Subtract 40 from both sides of the inequality to cancel out the constant term on the right side: 20 - 40 > -2u + 40 - 40 This simplifies to: -20 > -2u
02

- Solve for the variable

Divide both sides of the inequality by -2 to isolate the variable u. Remember that dividing by a negative number reverses the inequality sign: \[ \frac{-20}{-2} < \frac{-2u}{-2} \] which simplifies to: 10 < u
03

- Write the solution

Rewrite the solution to make it more standard and clear: u > 10
04

- Check the direction of the inequality sign

Since we divided both sides by a negative number, the inequality sign was correctly reversed. The final solution u > 10 is in the correct direction.

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Key Concepts

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

Isolating the Variable
To solve an inequality, your first goal is to isolate the variable you're interested in鈥攊n this case, the variable is \( u \). Isolating means getting \( u \) by itself on one side of the inequality.

In the problem \( 20 > -2u + 40 \), we need to remove the constant term (40) that's on the same side as our variable term (-2u). To do this, we subtract 40 from both sides of the inequality. By doing so, we're employing the concept of performing the same operation on both sides to maintain equality (or inequality in this case). Here鈥檚 how it looks:
  • Subtract 40 from both sides: \( 20 - 40 > -2u + 40 - 40 \)
  • Simplify: \( -20 > -2u \)
Great, now the variable term \( -2u \) is alone on one side of the inequality.
Reversing the Inequality Sign
In inequalities, one critical rule to remember is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This rule ensures the inequality remains true even when the orientation of the inequality changes.

Let's take our simplified inequality \( -20 > -2u \). To isolate \( u \) completely, we need to divide both sides by \( -2 \), a negative number.
  • Divide both sides by \(-2\): \(\frac{-20}{-2} < \frac{-2u}{-2}\)
  • Simplify: \( 10 < u \)
Notice how the '>' sign flipped to '<' when we divided by the negative number. This step ensures our solution remains accurate.
Checking Solutions
After solving an inequality, it's essential to verify the solution to ensure it's correct. This step can also help you understand the solution better.

For our problem, we concluded that the solution is \( u > 10 \). Let's check it.
  • Pick a value for \( u \) that is greater than 10, for example, 11.
  • Place \( u = 11 \) back into the original inequality: \( 20 > -2*11 + 40 \)
  • Simplify: \( 20 > -22 + 40 \)
  • Simplify further: \( 20 > 18 \)
Since 20 is indeed greater than 18, our solution \( u > 10 \) is correct and verifies that we handled the inequality sign reversal correctly.

Checking solutions not only confirms correctness but also solidifies your understanding of the problem and the rules applied.

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Most popular questions from this chapter

Describe the difference in the solution of \(x<9\) and the solution of \(x \leq 9\).

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