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Chapter 2: Problem 23

Solve the inequality. Then graph the solution set. $$3 x^{2}-11 x>20$$

Short Answer

Expert verified
The solution to the inequality \(3x^2 - 11x > 20\) is \((-鈭�, -4/3) 鈭� (5, 鈭�)\).

Step by step solution

01

Simplify

First, transfer the '20' from the right side to the left side of the inequality so that we have all the terms on one side. This results in the quadratic inequality: \(3x^2 - 11x - 20 > 0\)
02

Factorize

Next, we need to factor the quadratic. The factors of -60 (which is 3*20) that will sum to -11 are -15 and 4. Hence, the inequality can be written in the form of two binomials: \((3x + 4)(x - 5) > 0\)
03

Determine critical values

The roots of the equation, \(3x +4 = 0\) and \(x - 5 = 0\), gives us the critical points for the solution of the inequality. Solving them gives \(x= -4/3\) and \(x=5\).
04

Check the intervals

The intervals are \((-鈭�, -4/3)\), \((-4/3, 5)\), and \((5, 鈭�)\). Select a number from each interval and substitute it into the factored form of the equation to determine which interval(s) satisfy the original inequality. For this inequality, the solution is \((-鈭�, -4/3) 鈭� (5, 鈭�)\).
05

Draw the graph

Plot the critical points \(x= -4/3\) and \(x=5\) on a number line. Because the inequality is 'greater than but not equal to,' use open circles for the critical points. Then, draw a line to the right of \(x=5\) and to the left of \(x=-4/3\)

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