**Graph and Solve Quadratic Inequalities Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on graphing and solving quadratic inequalities.

**Problem 1 :**

Graph and solve for x :

x² - 7x + 6 > 0

**Problem 2 : **

Graph and solve for x :

-x² + 3x - 2 > 0

**Problem 3 :**

Graph and solve for x :

4x² - 25 ≥ 0

**Problem 4 :**

Graph and solve for x :

2x² − 12x + 50 ≤ 0

**Problem 1 :**

Graph and solve for x :

x² - 7x + 6 > 0

**Solution :**

x² - 7x + 6 > 0

(x - 1)(x - 6) > 0

On equating the factors to zero, we see that x = 1, x = 6 are the roots of the quadratic equation. Plotting these roots on real line and marking positive and negative alternatively from the right most part we obtain the corresponding number line as

If we plot these points on the number line, we will get intervals (-∞, 1) (1, 6) (6, ∞).

From (-∞, 1) let us take -1 |
(-1 − 1) (-1 − 6) > 0 -2(-7) > 0 14 > 0 |

From (1, 6) let us take 4 |
(4 − 1) (4 − 6) > 0 3(-2) > 0 -6 > 0 |

From (6, ∞) let us take 7 |
(7 − 1) (7 − 6) > 0 6(1) > 0 6 > 0 |

Hence, the solution set is

(− ∞, 1) ∪ (6, ∞)

**Problem 2 : **

Graph and solve for x :

-x² + 3x - 2 > 0

**Solution :**

-x² + 3x - 2 > 0

Multiplying by negative sign on both sides

x²- 3x + 2 < 0

(x − 1) (x − 2) < 0

x - 1 = 0 x - 2 = 0

x = 1 and x = 2

On equating the factors to zero, we see that x = 1, x = 2 are the roots of the quadratic equation. Plotting these roots on real line and marking positive and negative alternatively from the right most part we obtain the corresponding number line as

If we plot these points on the number line, we will get intervals (-∞, 1) (1, 2) (2, ∞).

From (-∞, 1) let us take -1 |
(x − 1) (x − 2) < 0 (-1 − 1) (-1 − 2) < 0 (-2)(-3) > 0 6 < 0 |

From (1, 2) let us take 1.5 |
(x − 1) (x − 2) < 0 (1.5 − 1) (1.5 − 2) < 0 (-0.5)(-0.5) > 0 0.25 < 0 |

From (2, ∞) let us take 7 |
(7 − 1) (7 − 6) > 0 6(1) > 0 6 > 0 False |

**Hence, the solution set is **

**(1, 2)**

**Problem 3 :**

Graph and solve for x :

4x² - 25 ≥ 0

**Solution :**

4x² - 25 ≥ 0

(2x)² - 5² ≥ 0

(2x - 5) (2x + 5) ≥ 0

2x - 5 ≥ 0 (or) 2x + 5 ≥ 0

x = 5/2 (or) x = -5/2

On equating the factors to zero, we see that x = 5/2, x = -5/2 are the roots of the quadratic equation. Plotting these roots on real line and marking positive and negative alternatively from the right most part we obtain the corresponding number line as

If we plot these points on the number line, we will get intervals (-∞, -5/2) (-5/2, 5/2) (5/2, ∞).Graph solutions to quadratic inequalities

From (-∞, -5/2) let us take -3 |
(2x - 5) (2x + 5) ≥ 0 (-6 − 5) (-6 + 5) < 0 (-11)(-1) > 0 11 > 0 |

From (-5/2, 5/2) let us take 0 |
(2x - 5) (2x + 5) ≥ 0 (0− 5) (0 + 5) < 0 (-5)(5) > 0 -25 > 0 |

From (5/2, ∞) let us take 4 |
(2x - 5) (2x + 5) ≥ 0 (8 − 5) (8 + 5) < 0 (3)(13) > 0 39 > 0 |

**Hence, the solution set is **

(-∞, -5/2) U (5/2, ∞)

**Problem 4 :**

Graph and solve for x :

2x² − 12x + 50 ≤ 0

**Solution :**

2x² − 12x + 50 ≤ 0

2(x²− 6x + 25) ≤ 0

x² − 6x + 25 ≤ 0

(x²− 6x + 9) + 25 − 9 ≤ 0

(x − 3)² + 16 ≤ 0

This is not true for any real value of x.

So, the given quadratic inequality has no solution.

After having gone through the stuff given above, we hope that the students would have understood, how to graph and solve quadratic inequalities.

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