Math Problem Statement

Which of the following value of x do not satisfy the inequality (x^2 - 3x + 2 > 0) at all?

Solution

The problem in the image asks:

Which of the following value of xx do not satisfy the inequality x23x+2>0x^2 - 3x + 2 > 0 at all?

The options are: a) 1x21 \leq x \leq 2
b) 1x2-1 \leq x \leq -2
c) x2x \leq 2
d) 0x20 \leq x \geq 2

Solution:

We start by solving the quadratic inequality x23x+2>0x^2 - 3x + 2 > 0.

Step 1: Solve x23x+2=0x^2 - 3x + 2 = 0

Factoring the quadratic equation: x23x+2=(x1)(x2)=0x^2 - 3x + 2 = (x - 1)(x - 2) = 0

So, the roots of the equation are x=1x = 1 and x=2x = 2.

Step 2: Analyze the sign of x23x+2x^2 - 3x + 2

Now, we need to determine where the quadratic expression is positive. We will check the intervals defined by the roots:

  • For x<1x < 1, pick x=0x = 0:
    x23x+2=023(0)+2=2>0x^2 - 3x + 2 = 0^2 - 3(0) + 2 = 2 > 0
    So, the inequality is satisfied for x<1x < 1.

  • For 1<x<21 < x < 2, pick x=1.5x = 1.5:
    x23x+2=1.523(1.5)+2=2.254.5+2=0.25<0x^2 - 3x + 2 = 1.5^2 - 3(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25 < 0
    So, the inequality is not satisfied for 1<x<21 < x < 2.

  • For x>2x > 2, pick x=3x = 3:
    x23x+2=323(3)+2=99+2=2>0x^2 - 3x + 2 = 3^2 - 3(3) + 2 = 9 - 9 + 2 = 2 > 0
    So, the inequality is satisfied for x>2x > 2.

Step 3: Combine results

The inequality x23x+2>0x^2 - 3x + 2 > 0 is satisfied for:

  • x<1x < 1
  • x>2x > 2

The inequality is not satisfied for 1x21 \leq x \leq 2.

Conclusion:

The correct answer is a) 1x21 \leq x \leq 2, as this interval does not satisfy the inequality at all.


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

Here are 5 related questions for further practice:

  1. Solve x25x+6>0x^2 - 5x + 6 > 0.
  2. Solve the inequality (x4)(x+2)>0(x - 4)(x + 2) > 0.
  3. Find the values of xx that satisfy x24x+3<0x^2 - 4x + 3 < 0.
  4. For what values of xx is x2+x60x^2 + x - 6 \geq 0?
  5. Solve the quadratic inequality 2x23x502x^2 - 3x - 5 \leq 0.

Tip: When solving quadratic inequalities, it helps to plot the parabola or analyze sign changes at the roots.

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Math Problem Analysis

Mathematical Concepts

Quadratic Inequality
Algebra
Factoring

Formulas

Quadratic equation formula: ax^2 + bx + c = 0

Theorems

Factorization of quadratic equations

Suitable Grade Level

Grades 8-10