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Chapter 2: Problem 15

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$4 x^{2}+7 x<-3$$

Short Answer

Expert verified
After performing above steps one would get the intervals which represent the solution of the given inequality. The exact solution may depend on the roots obtained from the quadratic formula in Step 2.

Step by step solution

01

Set the polynomial to equal zero

Solve the equation \(4x^2 + 7x + 3 = 0\) to find the critical values. These values divide the number line into regions where the inequality will hold true or not.
02

Solve for X

To solve the quadratic equation, use the quadratic formula \((-b 卤 \sqrt{b^2 - 4ac}) / (2a)\). Substitute \(a = 4\), \(b = 7\), and \(c = 3\) into the formula and solve for \(x\). After solving, you'll get two real roots for \(x\), which are the critical values.
03

Draw the number line

Draw a number line adding the critical values obtained from Step 2, these points will separate the number line into regions.
04

Choose test points and determine sign of inequality

Choose test points in each region on the number line and substitute them into the original inequality to determine whether the inequality is true or not. If the inequality is true for a test point in the region, then it is true for all points in that region.
05

Express the solution set in interval notation

Write down the solution set in interval notation. The solution is the union of all intervals where the inequality is true. Since this is a 'less than' inequality, you won't include the critical values (they will be represented with parenthesis).
06

Sketch the solution set on the number line

Draw the solution set on the number line. The solution intervals will be represented by solid or dashed lines depending on whether they include the end points or not.

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Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x-5}{10 x-2} \div \frac{x^{2}-10 x+25}{25 x^{2}-1}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000\). It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C,\) of producing \(x\) pairs of shoes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), \bar{C}(10,000),\) and \(\bar{C}(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this represents for the company.

Find the horizontal asymptote, if there is one, of the graph of rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{2 x+7}{x+3}$$
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