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Magazine Chapter 4: Problem 38
Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x+4}+x=8\) (b) \(\sqrt{3 x+4}+x>8\) (c) \(\sqrt{3 x+4}+x<8\)
Short Answer
Expert verified
Solution to (a): \(x = 4\). (b) \(x > 4\). (c) \(-\frac{4}{3} \leq x < 4\).
Step by step solution
01
Isolate the Square Root
To solve the equation \( \sqrt{3x+4} + x = 8 \), first isolate the square root term. Move \(x\) to the other side of the equation:\[ \sqrt{3x+4} = 8 - x \]
02
Square Both Sides
Next, square both sides of the equation to eliminate the square root:\[ (\sqrt{3x+4})^2 = (8-x)^2 \]This yields:\[ 3x+4 = (8-x)^2 \]
03
Expand and Simplify
Expand \((8-x)^2\):\[ 3x + 4 = 64 - 16x + x^2 \]Rearrange the equation to bring terms together:\[ x^2 - 19x + 60 = 0 \]
04
Solve the Quadratic Equation
This is a standard quadratic equation. Use the quadratic formula where \(a = 1\), \(b = -19\), and \(c = 60\).\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Calculate:\[ x = \frac{19 \pm \sqrt{(-19)^2 - 4 \times 1 \times 60}}{2 \times 1} \]\[ x = \frac{19 \pm \sqrt{361 - 240}}{2} \]\[ x = \frac{19 \pm \sqrt{121}}{2} \]\[ x = \frac{19 \pm 11}{2} \]This gives the solutions:\[ x = 15 \] or \[ x = 4 \]
05
Verify Solutions in the Original Equation
Substitute \(x = 15\) into the original equation:\[ \sqrt{3(15)+4} + 15 = \sqrt{49} + 15 = 7 + 15 = 22 eq 8 \]This is not valid.Substitute \(x = 4\) into the original equation:\[ \sqrt{3(4)+4} + 4 = \sqrt{16} + 4 = 4 + 4 = 8 \]This is valid.
06
Solve the Inequality for (b)
For the inequality \( \sqrt{3x+4} + x > 8 \), use the solutions above:Since \(x = 4\) satisfies the equality, \(x < 4\) makes the left side smaller than 8. Thus, the solution for the inequality is \(x > 4\). Verify the domain by ensuring the expression under the square root is non-negative: \(3x + 4 \geq 0\), which simplifies to \(x \geq -\frac{4}{3}\). So the solution is \(x > 4\).
07
Solve the Inequality for (c)
For the inequality \( \sqrt{3x+4} + x < 8 \), use the opposite range from the equality condition:Since \(x = 4\) is the boundary, \(x < 4\) makes the expression under consideration lesser than 8.The domain condition is \(x \geq -\frac{4}{3}\), so the solution is \(-\frac{4}{3} \leq x < 4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root Functions
Understanding square root functions is essential for solving equations involving them. A square root function is any function containing the square root of a variable. The general form is \( f(x) = \sqrt{x} \), but it can also be part of a more complex expression like \( \sqrt{3x+4} \).
Square root functions have unique properties:
Square root functions have unique properties:
- The domain of a square root function is limited to values where the expression inside the square root is non-negative. For example, in \( \sqrt{3x + 4} \), the domain is \( 3x + 4 \geq 0 \), which simplifies to \( x \geq -\frac{4}{3} \).
- Graphically, these functions typically start at the point where the expression inside the square root equals zero and rises slowly because they grow non-linear but continuous.
- They represent the principal (non-negative) root, meaning we only consider positive outputs unless otherwise specified.
Quadratic Equations
Quadratic equations appear in the form \( ax^2 + bx + c = 0 \), with solutions found using factoring, completing the square, or the quadratic formula. In our context, we derived a quadratic equation \( x^2 - 19x + 60 = 0 \) from squaring the equation \( \sqrt{3x+4} + x = 8 \).
Key aspects of solving quadratic equations include:
Key aspects of solving quadratic equations include:
- Identifying coefficients: Here, \( a = 1 \), \( b = -19 \), and \( c = 60 \).
- Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Applying this provides the roots \( x = 15 \) and \( x = 4 \).
- Verification: It's necessary to substitute solutions back into the original equation to ensure they work, as we did with \( x = 4 \).
Inequalities
Solving inequalities, like \( \sqrt{3x+4} + x > 8 \) or \( \sqrt{3x+4} + x < 8 \), involves determining the range of values that satisfy these conditions. Inequalities can be more complex than equations because they involve evaluating expressions over a range rather than at specific points.
Here's how to approach inequalities linked to our exercise:
Here's how to approach inequalities linked to our exercise:
- Identify the equality first, \( \sqrt{3x+4} + x = 8 \), which gives us the critical boundary, \( x = 4 \).
- For \( \sqrt{3x+4} + x > 8 \), solutions are \( x > 4 \) because any \( x \leq 4 \) makes the expression less than or equal to 8.
- For \( \sqrt{3x+4} + x < 8 \), solutions are \( -\frac{4}{3} \leq x < 4 \), ensuring the expression's non-negativity and remaining below 8.
