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Chapter 3: Problem 15

Solve each polynomial inequality in Exercises \(1-42\) 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
Solutions for given polynomial inequality are (-鈭�, -1) 鈭� (-3/4, 鈭�).

Step by step solution

01

Rewrite in Standard Form

First, the polynomial inequality \(4x^2 + 7x < -3\) can be rewritten as \(4x^2 + 7x + 3 > 0\). This is done by adding 3 on both sides.
02

Find the Roots

Next, we need to find the roots, meaning the values for which the polynomial equals zero. We can do this by applying the quadratic formula \(-\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\), where a=4, b=7, and c=3. This results in 鈭�1 and 鈭�3/4 as the roots.
03

Testing the Regions

Now we have three regions defined by the roots: (-鈭�, -1), (-1, -3/4), and (-3/4, 鈭�). To identify if an interval satisfies the inequality, 'test' values can be picked from each of the regions. If the inequality holds true, then the region is a part of the solution. In this case, the 'test' values could be -2, -0.875, and 0, resulting in the regions (-鈭�, -1) and (-3/4, 鈭�) satisfying the inequality.
04

Graph the Solution

The solution can now be graphed on a number line, indicating the solution intervals. The intervals (-鈭�, -1) and (-3/4, 鈭�) are part of the solution, demonstrated by marking the 'test' values on the number line.
05

Express the Solution in Interval Notation

Finally, the solution intervals can be stated in interval notation. Open intervals are used because the inequality does not include 'equal to' and the solutions are: (-鈭�, -1) 鈭� (-3/4, 鈭�).

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