Questions: Solve the compound inequality. 4x + 5 < 17 or 2x - 6 < 6 Write the solution in interval notation. If there is no solution, enter ∅.

Solve the compound inequality. 4x + 5 < 17 or 2x - 6 < 6 Write the solution in interval notation. If there is no solution, enter ∅.
Transcript text: Solve the compound inequality. \[ 4 x+5<17 \text { or } 2 x-6<6 \] Write the solution in interval notation. If there is no solution, enter $\varnothing$.
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Solution

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Solution Steps

Step 1: Solve the First Inequality

The first inequality is:

\[ 4x + 5 < 17 \]

Subtract 5 from both sides:

\[ 4x < 12 \]

Divide both sides by 4:

\[ x < 3 \]

Step 2: Solve the Second Inequality

The second inequality is:

\[ 2x - 6 < 6 \]

Add 6 to both sides:

\[ 2x < 12 \]

Divide both sides by 2:

\[ x < 6 \]

Step 3: Combine the Solutions

The compound inequality is an "or" statement, meaning we take the union of the solutions. The solutions are:

  • From the first inequality: \(x < 3\)
  • From the second inequality: \(x < 6\)

The union of these solutions is \(x < 6\), since any \(x\) that satisfies \(x < 3\) also satisfies \(x < 6\).

Final Answer

The solution in interval notation is:

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

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