Math Problem Statement

Solve the inequality x^2 + 7x ≥ 4.

Solution

The inequality in the image is:

x2+7x4x^2 + 7x \geq 4

Let's solve this inequality.

Step 1: Move all terms to one side of the inequality.

Subtract 4 from both sides:

x2+7x40x^2 + 7x - 4 \geq 0

Step 2: Solve the corresponding quadratic equation.

To find the critical points, solve the quadratic equation:

x2+7x4=0x^2 + 7x - 4 = 0

Use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=1a = 1, b=7b = 7, and c=4c = -4.

x=7±724(1)(4)2(1)x = \frac{-7 \pm \sqrt{7^2 - 4(1)(-4)}}{2(1)} x=7±49+162x = \frac{-7 \pm \sqrt{49 + 16}}{2} x=7±652x = \frac{-7 \pm \sqrt{65}}{2}

So the solutions are:

x=7+652andx=7652x = \frac{-7 + \sqrt{65}}{2} \quad \text{and} \quad x = \frac{-7 - \sqrt{65}}{2}

Step 3: Analyze the intervals.

The critical points divide the number line into three intervals. Test the sign of the quadratic expression x2+7x4x^2 + 7x - 4 in each interval to determine where the inequality holds.

  1. x<7652x < \frac{-7 - \sqrt{65}}{2}
  2. 7652x7+652\frac{-7 - \sqrt{65}}{2} \leq x \leq \frac{-7 + \sqrt{65}}{2}
  3. x>7+652x > \frac{-7 + \sqrt{65}}{2}

Step 4: Conclusion

  • The quadratic expression is greater than or equal to zero in intervals where the test values yield positive results.
  • Therefore, the solution to the inequality is:

x(,7652][7+652,)x \in \left(-\infty, \frac{-7 - \sqrt{65}}{2}\right] \cup \left[\frac{-7 + \sqrt{65}}{2}, \infty\right)

Would you like a more detailed explanation on any step or have any questions?


Here are 5 related questions that you may find helpful:

  1. How do you determine the sign of a quadratic expression in different intervals?
  2. What is the significance of the discriminant in the quadratic formula?
  3. Can this inequality be solved using a different method?
  4. How would the solution change if the inequality were strict (i.e., >> instead of \geq)?
  5. What are the applications of solving quadratic inequalities?

Tip: Always check the sign of a quadratic expression by testing values in each interval defined by the roots of the equation. This helps in accurately determining the solution set for inequalities.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Inequalities
Quadratic Equations

Formulas

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

Theorems

Sign analysis of a quadratic expression
Roots of a quadratic equation

Suitable Grade Level

Grades 9-12