5.5 Calculate the $$ x $$-coordinates of the points of intersection of $$ f $$ and $$ g $$. 5.6 If $$ x

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5.5 Calculate the $$ x $$-coordinates of the points of intersection of $$ f $$ and $$ g $$. 5.6 If $$ x<3 $$, determine the values of $$ x $$ for which $$ \frac{4}{x-3}+2

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To assist you effectively with problems **5.5**, **5.6**, and **5.7**, let's address each one individually. However, it's essential to note that problems **5.5** and **5.7** require specific information about the functions $$ f $$ and $$ g $$ which aren't provided in your query. I'll provide a detailed solution for **5.6** and guidance on what information is needed for the other parts. --- ### **Problem 5.6** **If $$ x < 3 $$, determine the values of $$ x $$ for which** $$ \frac{4}{x - 3} + 2 < x + 2 $$ **Solution:** 1. **Simplify the Inequality:** $$ \frac{4}{x - 3} + 2 < x + 2 $$ Subtract 2 from both sides to isolate the fraction: $$ \frac{4}{x - 3} < x $$ 2. **Understand the Domain:** The denominator $$ x - 3 $$ cannot be zero, so $$ x \neq 3 $$. Given that $$ x < 3 $$, the denominator $$ x - 3 $$ is negative. 3. **Solve the Inequality:** Multiply both sides by $$ x - 3 $$. **Note:** Since $$ x - 3 < 0 $$, multiplying both sides by it will reverse the inequality sign. $$ 4 > x(x - 3) $$ Simplify the right side: $$ 4 > x^2 - 3x $$ Rearrange the inequality: $$ x^2 - 3x - 4 < 0 $$ 4. **Find the Roots of the Quadratic:** Solve $$ x^2 - 3x - 4 = 0 $$: $$ x = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} $$ So, the roots are: $$ x = 4 \quad \text{and} \quad x = -1 $$ 5. **Determine the Intervals:** The quadratic $$ x^2 - 3x - 4 $$ is less than 0 between its roots: $$ -1 < x < 4 $$ 6. **Apply the Given Condition $$ x < 3 $$:** Combine the above interval with $$ x < 3 $$: $$ -1 < x < 3 $$ **Final Answer for Problem 5.6:** $$ x \in (-1,\, 3) $$ --- ### **Problems 5.5 and 5.7** **Issues:** - **Problem 5.5:** *Calculate the $$ x $$-coordinates of the points of intersection of $$ f $$ and $$ g $$.* - **Problem 5.7:** *The line $$ y = x - 1 $$ cuts $$ f $$ at $$ \mathrm{P}(1 ; 0) $$ and Q. Write down the coordinates of Q.* **What’s Needed:** To solve these problems, the explicit definitions or equations of the functions $$ f(x) $$ and $$ g(x) $$ are required. Without knowing what $$ f $$ and $$ g $$ represent (e.g., linear functions, quadratics, etc.), it's impossible to determine their points of intersection or the coordinates of point $$ Q $$. **Guidance:** - **For Problem 5.5:** Once you have the equations for $$ f(x) $$ and $$ g(x) $$, set them equal to each other and solve for $$ x $$. The solutions will give the $$ x $$-coordinates of the points where the two functions intersect. $$ f(x) = g(x) \quad \Rightarrow \quad x\text{-coordinates of intersections} $$ - **For Problem 5.7:** Knowing $$ f(x) $$, substitute $$ y = x - 1 $$ into $$ f(x) $$ to find the points of intersection. Since one point $$ P(1, 0) $$ is given, solving the equation will yield the coordinates of the second point $$ Q $$. $$ f(x) = x - 1 \quad \Rightarrow \quad \text{Solve for } x \text{ to find Q.} $$ --- **If you can provide the equations for $$ f $$ and $$ g $$, I’d be happy to help you solve Problems 5.5 and 5.7 in detail!** Cannot determine the solutions for problems 5.5 and 5.7 without additional information about the functions $$ f $$ and $$ g $$.

5.5 Calculate the \( x \)-coordinates of the points of intersection of \( f \) and \( g \). 5.6 If \( x<3 \), determine the values of \( x \) for which \( \frac{4}{x-3}+2

Ask by Hardy Cole. in South Africa

Feb 03,2025

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5.5 Calculate the \( x \)-coordinates of the points of intersection of \( f \) and \( g \). 5.6 If \( x<3 \), determine the values of \( x \) for which \( \frac{4}{x-3}+2 Ask by Hardy Cole. in South Africa Feb 03,2025