Math Problem Statement

Solve this by factoring

Solution

To solve the quadratic inequality 4x2+7x204x^2 + 7x - 2 \geq 0 by factoring, we need to follow these steps:

1. Factor the quadratic expression

We need to find two numbers that multiply to 4×(2)=84 \times (-2) = -8 and add to 77. The numbers 88 and 1-1 fit these conditions. Thus, we rewrite the middle term:

4x2+8xx204x^2 + 8x - x - 2 \geq 0

Next, we group and factor:

(4x2+8x)(x+2)0(4x^2 + 8x) - (x + 2) \geq 0

Factor each group:

4x(x+2)1(x+2)04x(x + 2) - 1(x + 2) \geq 0

Now factor out the common factor:

(4x1)(x+2)0(4x - 1)(x + 2) \geq 0

2. Determine the critical points

Set each factor equal to zero to find the critical points:

4x1=0x=144x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{4} x+2=0x=2x + 2 = 0 \quad \Rightarrow \quad x = -2

These critical points divide the number line into intervals: (,2)(- \infty, -2), (2,14)(-2, \frac{1}{4}), and (14,)(\frac{1}{4}, \infty).

3. Test the intervals

We will test the sign of the expression (4x1)(x+2)(4x - 1)(x + 2) in each interval.

  • For x(,2)x \in (-\infty, -2), choose x=3x = -3: (4(3)1)((3)+2)=(13)(1)>0(4(-3) - 1)((-3) + 2) = (-13)(-1) > 0

  • For x(2,14)x \in (-2, \frac{1}{4}), choose x=0x = 0: (4(0)1)((0)+2)=(1)(2)<0(4(0) - 1)((0) + 2) = (-1)(2) < 0

  • For x(14,)x \in (\frac{1}{4}, \infty), choose x=1x = 1: (4(1)1)(1+2)=(3)(3)>0(4(1) - 1)(1 + 2) = (3)(3) > 0

4. Write the solution in interval notation

The inequality is satisfied in the intervals where the product is non-negative, which are (,2](- \infty, -2] and [14,)[\frac{1}{4}, \infty).

5. Set-builder notation

The solution set in set-builder notation is:

{xx2 or x14}\{x \mid x \leq -2 \text{ or } x \geq \frac{1}{4}\}

Would you like more details or have any questions?

Here are some related questions you might find helpful:

  1. How do you determine which intervals to test for signs in a quadratic inequality?
  2. Can all quadratic inequalities be solved by factoring?
  3. What if the quadratic expression is not factorable? How should you proceed?
  4. How do the signs of the factors influence the solution of an inequality?
  5. Why is it important to include the critical points when the inequality is non-strict (e.g., 0\geq 0)?

Tip: Always double-check your factorization and the intervals you test to ensure accuracy in solving inequalities.

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Math Problem Analysis

Mathematical Concepts

Quadratic Inequalities
Factoring

Formulas

Quadratic inequality formula ax^2 + bx + c ≥ 0

Theorems

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Suitable Grade Level

Grades 9-12