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Chapter 4: Problem 30

Solve. $$x^{2}+4 x+7 \geq 5 x+9$$

Short Answer

Expert verified
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Step by step solution

01

Move all terms to one side

Subtract 5x and 9 from both sides of the inequality to set it to 0: \[ x^2 + 4x + 7 - 5x - 9 \ x^2 - x - 2 \ x^2 - x - 2 \ x^2 - x - 2 \ x^2 - x - 2 \ x^2 - x 2\leq 0 \]
02

:4032]: -Factor the quad tor =0 get the making terms

The equre on faccommate fausing the form ( . \[ (x-2)(x-1) = 0 \] \ text here

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving inequalities
To solve quadratic inequalities, such as the given problem, we need to follow a series of logical steps. Solving inequalities means finding the range of values for the variable that make the inequality true.
In this problem, we start with the inequality: \(x^{2}+4x+7 \geq 5x+9\). The goal is to isolate the variable on one side.
This involves moving terms to one side to create a standard quadratic inequality format. Once we have it in the form \(ax^2 + bx + c \leq 0\), we are ready to determine the solution range.
factoring quadratics
Factoring is crucial in solving quadratic inequalities. It involves expressing the quadratic equation as a product of linear factors.
For the inequality \(x^2 - x - 2 \leq 0\), we factor it into two binomials.
The process of factoring involves finding two numbers that multiply to give the constant term (here -2) and add up to the coefficient of the middle term (here -1).
In this case, the factors are (x-2)(x+1).
Checking our factors: \( (x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2 \). Therefore, the quadratic factors correctly.
polynomial functions
In the context of inequalities, recognizing the shape of the polynomial function helps in understanding the solutions.
The polynomial \( x^2 - x - 2 \) is a parabola that opens upwards because the coefficient of \( x^2 \) is positive.
The roots, or x-intercepts, are found using the factors: x = 2 and x = -1.
These roots divide the x-axis into intervals.
By choosing test points from each interval and substituting them into the factored inequality \((x-2)(x+1)\), we determine which intervals satisfy the inequality.
step-by-step solution
Here's a detailed step-by-step solution to the given problem:
1. Move all terms to one side: \(x^2 + 4x + 7 - 5x - 9 \geq 0\) simplifies to \(x^2 - x - 2 \leq 0\).
2. Factor the quadratic: \(x^2 - x - 2 = (x-2)(x+1)\).
3. Find the roots: Set each factor to zero, \((x-2) = 0\) gives x = 2, and \((x+1) = 0\) gives x = -1.
4. Determine intervals: The roots divide the number line into intervals: \((-fty, -1)\), \((-1,2)\), and \((2, fty)\).
5. Test these intervals in the inequality \( (x-2)(x+1) \leq 0 \). Choose sample points, e.g., x = -2, x = 0, and x = 3 to check which intervals satisfy the inequality.
6. The solution is x in the interval \([-1, 2]\).

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Most popular questions from this chapter

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the \(x\) -intercepts and the \(y\) -intercepts. Check your work using \(a\) graphing calculator. $$f(x)=\frac{x^{2}}{x^{2}-x-2}$$

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