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Use deductive reasoning to prove that [tex]$x = -3$[/tex] is not in the solution set of the inequality [tex]-4x \ \textless \ 8[/tex]. Be sure to justify each step in your proof.
Sure, let's go through the steps to prove whether [tex]\( x = -3 \)[/tex] satisfies the inequality [tex]\( -4x < 8 \)[/tex].
### Step-by-Step Solution:
1. Substitute [tex]\( x = -3 \)[/tex] into the inequality: - The given inequality is [tex]\( -4x < 8 \)[/tex]. - Substitute [tex]\( x = -3 \)[/tex] into the left-hand side of the inequality: [tex]\[
-4(-3)
\][/tex]
2. Simplify the left-hand side: - When we multiply [tex]\(-4\)[/tex] by [tex]\(-3\)[/tex], we get: [tex]\[
-4 \times (-3) = 12
\][/tex]
3. Evaluate the inequality: - Substitute [tex]\( 12 \)[/tex] back in for the left-hand side: [tex]\[
12 < 8
\][/tex] - Now, let's evaluate whether this statement is true or false.
4. Check the truth of the statement: - Clearly, [tex]\( 12 \)[/tex] is not less than [tex]\( 8 \)[/tex]. [tex]\[
12 \geq 8
\][/tex] - Therefore, [tex]\( 12 < 8 \)[/tex] is a false statement.
### Conclusion: Since the statement [tex]\( 12 < 8 \)[/tex] is false, it means that substituting [tex]\( x = -3 \)[/tex] into the inequality [tex]\( -4x < 8 \)[/tex] does not satisfy the inequality. Thus, [tex]\( x = -3 \)[/tex] is not in the solution set of the inequality [tex]\( -4x < 8 \)[/tex].
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