65) If $$ 3 x+7 \geq x+5 \geq 5 x-3 $$, then the inequality holds when $$ x $$ lies in the range \begin{tabular}{|l|l}\hline a) $$ 7 \leq x \leq-3 $$ & b) $$ 5 \geq x \geq 7 $$ \ \hline c) $$ 3 \leq x \leq 5 $$ & d) $$ 2 \geq x \geq-1 $$ \ 66) If $$ 3 x+7

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65) If $$ 3 x+7 \geq x+5 \geq 5 x-3 $$, then the inequality holds when $$ x $$ lies in the range \begin{tabular}{|l|l}\hline a) $$ 7 \leq x \leq-3 $$ & b) $$ 5 \geq x \geq 7 $$ \ \hline c) $$ 3 \leq x \leq 5 $$ & d) $$ 2 \geq x \geq-1 $$ \ 66) If $$ 3 x+7<5 x-3 $$ then the ineaualitv holds when $$ x $$ lies in the range \end{tabular}

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Let's solve each of the inequalities step by step. --- ### **Problem 65** **Given:** $$ 3x + 7 \geq x + 5 \geq 5x - 3 $$ This is a **double inequality**, which can be broken down into two separate inequalities: 1. **First Inequality:** $$ 3x + 7 \geq x + 5 $$ **Solving:** $$ \begin{align*} 3x + 7 &\geq x + 5 \ 3x - x &\geq 5 - 7 \ 2x &\geq -2 \ x &\geq -1 \ \end{align*} $$ 2. **Second Inequality:** $$ x + 5 \geq 5x - 3 $$ **Solving:** $$ \begin{align*} x + 5 &\geq 5x - 3 \ 5 + 3 &\geq 5x - x \ 8 &\geq 4x \ x &\leq 2 \ \end{align*} $$ **Combined Solution:** $$ -1 \leq x \leq 2 $$ **Matching with Options:** - **Option d)** $$ 2 \geq x \geq -1 $$ matches the solution. --- ### **Problem 66** **Given:** $$ 3x + 7 < 5x - 3 $$ **Solving the Inequality:** $$ \begin{align*} 3x + 7 &< 5x - 3 \ 7 + 3 &< 5x - 3x \ 10 &< 2x \ x &> 5 \ \end{align*} $$ **Solution:** $$ x > 5 $$ --- **Final Answers:** - **Problem 65 Answer:** Option **d)** $$2 \geq x \geq -1$$. - **Problem 66 Solution:** The inequality holds for all real numbers $$x$$ with $$x > 5$$. **Problem 65 Answer:** Option **d)** $$2 \geq x \geq -1$$. **Problem 66 Solution:** $$x > 5$$.

If , then the inequality holds when lies in the range

a) b)

c) d)

66) If then the ineaualitv holds when lies in the range

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66) If then the ineaualitv holds when lies in the range

If , then the inequality holds when lies in the range a) b) c) d) 66) If then the ineaualitv holds when lies in the range