Which inequality describes all solutions to 5(3 – x) < –2x + 6

To solve the inequality \(5(3 - x) < -2x + 6\), let's break it down step by step:

### Step 1: Expand the left side of the inequality.
Distribute the 5 across the terms inside the parentheses:

\[
5(3 - x) = 5 \cdot 3 - 5 \cdot x = 15 - 5x
\]

Now the inequality is:

\[
15 - 5x < -2x + 6
\]

### Step 2: Move all terms involving \(x\) to one side.
Add \(5x\) to both sides:

\[
15 < 5x - 2x + 6
\]

Simplify:

\[
15 < 3x + 6
\]

### Step 3: Move the constant terms to one side.
Subtract 6 from both sides:

\[
15 - 6 < 3x
\]

Simplify:

\[
9 < 3x
\]

### Step 4: Solve for \(x\).
Divide both sides by 3:

\[
x > 3
\]

### Final Solution:
The inequality that describes all solutions is:

\[
x > 3
\]

Would you like further clarification on any of the steps?

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Here are some related questions to consider:
1. How would the solution change if the inequality was \(5(3 - x) \leq -2x + 6\)?
2. Can you graph the solution \(x > 3\) on a number line?
3. What is the significance of flipping the inequality sign when multiplying or dividing by a negative number?
4. How would you solve if there were fractions involved in the inequality?
5. How do you solve a compound inequality, such as \(5(3 - x) < -2x + 6 < 10\)?

**Tip:** Always check your solution by substituting values back into the original inequality!

Which inequality describes all solutions to 5(3 – x) < –2x + 6?

Learn how to solve the linear inequality 5(3 - x) < -2x + 6 step by step. This solution covers essential algebraic concepts, including the distributive property and properties of inequalities. Ideal for students in Grades 8-10.

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