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Chapter 7: Problem 106

Solve. $$10-5 y>4$$

Short Answer

Expert verified
y < 6/5

Step by step solution

01

- Subtract 10 from both sides

To isolate the term with the variable, subtract 10 from both sides of the inequality: 10 - 10 - 5y > 4 - 10 Simplifying, we get: -5y > -6
02

- Divide by -5

Since we have a negative coefficient for y, we need to divide both sides by -5. Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips: -5y / -5 < -6 / -5 y < 6/5

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Key Concepts

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

Solving Inequalities
Solving inequalities is similar to solving equations, but there are extra rules to consider. For this exercise, we start with the inequality:\(10 - 5y > 4\)The goal is to isolate the variable on one side of the inequality. We'll begin by subtracting 10 from both sides, making sure to simplify as we go. The key is to perform the same operation on each side of the inequality to keep it balanced.
  • Subtract 10: \(10 - 10 - 5y > 4 - 10\)
  • Simplify: \(-5y > -6\)
Once we have isolated the term with the variable, we move on to the next step.
Linear Inequalities
A linear inequality looks very similar to a linear equation, but instead of an equals sign, it uses inequality signs such as \(>\), \(<\), \(\geq\), or \(\leq\). In this problem, we started with \(10 - 5y > 4\), which is a common form of linear inequality.To solve a linear inequality, follow similar steps to solving linear equations:
  • Isolate the variable term on one side
  • Perform inverse operations such as addition, subtraction, multiplication, or division
  • Simplify the inequality
In our example, after isolating and simplifying the term with y, we ended up with \(-5y > -6\).
Inequality Rules
When working with inequalities, certain rules are crucial. One important rule applies when multiplying or dividing both sides of an inequality by a negative number. This action flips the inequality sign.In our exercise, we had:\(-5y > -6\)To solve for \(y\), we divided both sides by -5. Note that we flipped the \(>\) sign to \(<\) during this process, leading to:\(y < \frac{6}{5}\)Here's a quick summary of important inequality rules:
  • When adding or subtracting the same number from both sides, the inequality sign remains the same.
  • When multiplying or dividing both sides by a positive number, the inequality sign remains the same.
  • When multiplying or dividing both sides by a negative number, flip the inequality sign.
Remembering these rules helps ensure that the inequality solutions are accurate.

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Most popular questions from this chapter

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt{a b^{3}}}{\sqrt[5]{a^{2} b^{3}}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[3]{x^{2}}}{\sqrt[5]{x}} $$

Multiply. $$ (\sqrt{x+2}-\sqrt{x-2})^{2} $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=x-\sqrt{2}, g(x)=x+\sqrt{6} $$

Simplify. $$ 7 x \sqrt{(x+y)^{3}}-5 x y \sqrt{x+y}-2 y \sqrt{(x+y)^{3}} $$
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