Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 4

Chapter 4, Problem 4

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. (x+1)(x−7)≤0

Verified step by step guidance

Start by identifying the critical points of the inequality by setting each factor equal to zero: solve $x + 1 = 0$ and $x - 7 = 0$ to find the values of $x$ where the expression changes sign.

The critical points divide the real number line into intervals. These intervals are $(-\infty, -1)$, $[-1, 7]$, and $(7, \infty)$. We will test each interval to determine where the inequality $(x+1)(x-7) \leq 0$ holds true.

Choose a test point from each interval and substitute it into the expression $(x+1)(x-7)$. Check whether the product is less than or equal to zero for that interval.

Based on the sign of the product in each interval, determine which intervals satisfy the inequality. Remember to include the points where the product equals zero because the inequality is 'less than or equal to zero'.

Express the solution set using interval notation, combining all intervals where the inequality holds, and then graph this solution set on the real number line by shading the appropriate regions and marking the critical points.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are values of the variable where the polynomial equals zero, dividing the number line into intervals. By testing points in each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds.

Interval Notation and Graphing on the Number Line

Interval notation expresses solution sets as ranges of values, using parentheses for strict inequalities and brackets for inclusive ones. Graphing on the number line visually represents these intervals, showing where the solution lies and whether endpoints are included.

Recommended video:

05:18

Interval Notation

Related Practice

Textbook Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f (x) = - x^{2} + 2 x + 3$

views

Textbook Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x⁴−11x³−x²+19x+6

views

Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. $h (x) = 7 x^{3} + 2 x^{2} + \frac{1}{x}$

views

Textbook Question

In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. $f (x) = - x^{2} + 14 x - 106$

views

Textbook Question

Divide using long division. State the quotient, and the remainder, r(x). (x³+5x²+7x+2)÷(x+2)

views

Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x⁷+πx⁵+2/3 x