Math Problem Statement

Find the solution of the inequality 4x^2 + 7x - 2 ≥ 0 and express the solution in both Set-Builder Notation and Interval Notation.

Solution

The problem in the image requires finding the solution to the inequality 4x2+7x204x^2 + 7x - 2 \geq 0 and then expressing the solution in both Set-Builder Notation and Interval Notation.

Step 1: Solve the inequality 4x2+7x204x^2 + 7x - 2 \geq 0

We first need to solve the corresponding quadratic equation: 4x2+7x2=04x^2 + 7x - 2 = 0 We can solve this quadratic equation using the quadratic formula: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=4a = 4, b=7b = 7, and c=2c = -2.

x=7±724(4)(2)2(4)x = \frac{-7 \pm \sqrt{7^2 - 4(4)(-2)}}{2(4)} x=7±49+328x = \frac{-7 \pm \sqrt{49 + 32}}{8} x=7±818x = \frac{-7 \pm \sqrt{81}}{8} x=7±98x = \frac{-7 \pm 9}{8} So, we have two solutions: x1=7+98=28=14x_1 = \frac{-7 + 9}{8} = \frac{2}{8} = \frac{1}{4} x2=798=168=2x_2 = \frac{-7 - 9}{8} = \frac{-16}{8} = -2

Step 2: Determine the sign of 4x2+7x24x^2 + 7x - 2

The quadratic opens upwards because the coefficient of x2x^2 is positive. The roots divide the number line into three intervals:

  1. (,2)(-\infty, -2)
  2. (2,14)(-2, \frac{1}{4})
  3. (14,)(\frac{1}{4}, \infty)

We test the sign of the quadratic expression in each interval:

  • For x(,2)x \in (-\infty, -2), pick x=3x = -3: 4(3)2+7(3)2=36212=13>04(-3)^2 + 7(-3) - 2 = 36 - 21 - 2 = 13 > 0
  • For x(2,14)x \in (-2, \frac{1}{4}), pick x=0x = 0: 4(0)2+7(0)2=2<04(0)^2 + 7(0) - 2 = -2 < 0
  • For x(14,)x \in (\frac{1}{4}, \infty), pick x=1x = 1: 4(1)2+7(1)2=4+72=9>04(1)^2 + 7(1) - 2 = 4 + 7 - 2 = 9 > 0

So, the quadratic expression is positive (or zero) on the intervals (,2](-\infty, -2] and [14,)[\frac{1}{4}, \infty).

Step 3: Write the solution in Set-Builder and Interval Notation

  • Set-Builder Notation: {xx2 or x14}\{ x \,|\, x \leq -2 \text{ or } x \geq \frac{1}{4} \}

  • Interval Notation: (,2][14,)(-\infty, -2] \cup \left[\frac{1}{4}, \infty\right)

Would you like more details on any step? Here are some related questions:

  1. How do you determine the sign of a quadratic expression in different intervals?
  2. What is the significance of the direction in which a parabola opens?
  3. How does the quadratic formula work, and why is it applicable here?
  4. Can we solve the quadratic inequality using a different method, like factoring?
  5. What is the difference between Set-Builder Notation and Interval Notation?

Tip: When solving inequalities, always consider the intervals between the roots and test them to understand the behavior of the quadratic function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Inequalities
Set-Builder Notation
Interval Notation

Formulas

Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Quadratic Formula
Sign Analysis for Quadratic Expressions

Suitable Grade Level

Grades 9-11