Questions: Solve the inequality. 2(x+4)-8>-4(x-3)+24 The solution set is (Type your answer in interval notation)

Solve the inequality. 2(x+4)-8>-4(x-3)+24 The solution set is (Type your answer in interval notation)
Transcript text: Solve the inequality. \[ 2(x+4)-8>-4(x-3)+24 \] The solution set is $\square$ (Type your answer in interval notation )
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Solution

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Solution Steps

To solve the inequality \(2(x+4)-8 > -4(x-3)+24\), we will first expand and simplify both sides of the inequality. Then, we will collect all terms involving \(x\) on one side and constant terms on the other side. Finally, we will solve for \(x\) and express the solution in interval notation.

Step 1: Distribute and Simplify Both Sides

First, distribute the constants in the expressions on both sides of the inequality:

\[ 2(x+4) - 8 > -4(x-3) + 24 \]

Distribute the 2 on the left side:

\[ 2x + 8 - 8 > -4(x-3) + 24 \]

Simplify the left side:

\[ 2x > -4(x-3) + 24 \]

Distribute the -4 on the right side:

\[ 2x > -4x + 12 + 24 \]

Simplify the right side:

\[ 2x > -4x + 36 \]

Step 2: Combine Like Terms

Add \(4x\) to both sides to get all the \(x\) terms on one side:

\[ 2x + 4x > 36 \]

Combine the \(x\) terms:

\[ 6x > 36 \]

Step 3: Solve for \(x\)

Divide both sides by 6 to solve for \(x\):

\[ x > \frac{36}{6} \]

Simplify the division:

\[ x > 6 \]

Final Answer

The solution set in interval notation is:

\[ \boxed{(6, \infty)} \]

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