Solving Quadratic Inequalities – Explanation
& Examples
To solve a quadratic inequality, we also apply the same method as illustrated in the
procedure below:
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Write the quadratic inequality in standard form: ax2 + bx + c where a, b and are
coefficients and a ≠ 0
Determine the roots of the inequality.
Write the solution in inequality notation or interval notation.
If the quadratic inequality is in the form: (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, and if it is
in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.
Example 1
Solve the inequality x2 – 4x > –3
Solution
First, make one side one side of the inequality zero by adding both sides by 3.
x2 – 4x > –3 ⟹ x2 – 4x + 3 > 0
Factor the left side of the inequality.
x2 – 4x + 3 > 0 ⟹ (x – 3) (x – 1) > 0
Solve for all the zeroes for the inequality;
For, (x – 1) > 0 ⟹ x > 1 and for, (x – 3) > 0 ⟹ x>3
Since y is positive, we therefore choose the values of x which the curve will be above
the x-axis.
x < 1 or x > 3
Example 2
Solve the inequality x2 – x > 12.
Solution
To write the inequality in standard form, subtract both sides of the inequality by 12.
x2 – x > 12 ⟹ x2 – x – 12 > 0.
Factorize the quadratic inequality to get to;
(x – 4) (x + 3) > 0
Solve for all the zeroes for the inequality;
For, (x + 3) > 0 ⟹ x > -3
For x – 4 > 0 ⟹ x > 4
The values x < –3 or x > 4 are therefore the solution of this quadratic inequality.
Example 3
Solve 2x2 < 9x + 5
Solution
Write the inequality in standard form by making one side of the inequality zero.
2x2 < 9x + 5 ⟹ 2x2 – 9x – 5 < 0
Factor the left side of the quadratic inequality.
2x2 – 9x – 5 < 0 ⟹ (2x + 1) (x – 5) < 0
Solve for all the zeroes for the inequality
For, (x – 5) < 0 ⟹ x < 5 and for (2x + 1) < 0 ⟹ x < -1/2
Since y is negative for the equation 2x2 – 9x – 5 < 0, we therefore choose the values of
x which the curve will be below the x axis.
Therefore, the solution is -1/2 < x < 5
Example 4
Solve – x 2 + 4 < 0. Solution
Since the inequality is already in standard form, we therefore factor the expression.
-x 2 + 4 < 0 ⟹ (x + 2) (x – 2) < 0
Solve for all the zeroes for the inequality
For, (x + 2) < 0 ⟹ x < -2 and for, (x – 2) < 0 ⟹ x < 2
The y for –x 2 + 4 < 0 is negative; therefore, we choose the values of x in which the
curve will below the x- axis: –2 < x > 2
Example 5
Solve 2x2 + x − 15 ≤ 0.
Solution
Factor the quadratic equation.
2x2 + x − 15 = 0
2x2 + 6x – 5x− 15 = 0
2x (x + 3) – 5(x + 3) = 0
(2x – 5) (x + 3) = 0
For, 2x – 5 = 0 ⟹ x= 5/2 and for, x + 3= 0 ⟹ x = -3
Since the y for 2x2 + x − 15 ≤ 0 is negative, the we choose the values of x in which the
curve will be below the x axis. Therefore, x ≤ -3 or x ≥5/2 is the solution.
Example 6
Solve – x2 + 3x − 2 ≥ 0
Solution
Multiply the quadratic equation by -1 and remember to change the sign.
x2 – 3x + 2 = 0
x2 – 1x – 2x + 2 = 0
x (x – 1) – 2(x – 1) = 0
(x – 2) (x – 1) = 0
For, x – 2 = 0 ⟹ x = 2 and for, x – 1= 0 ⟹x=1
Therefore, the solution to the quadratic inequality is 1 ≤ x ≤ 2
Example 7
Solve x2 − 3x + 2 > 0
Solution
Factorize the expression to get;
x2 − 3x + 2 > 0 ⟹ (x − 2) (x − 1) > 0
Now solve for the roots of the inequality as;
(x − 2) > 0 ⟹ x > 2
(x − 1) > 0 ⟹x > 1
The curve for x2 − 3x + 2 > 0 has positive y, therefore which choose the values of x in which the
curve will be above the x-axis. The solution is hence, x < 1 or x > 2.
Example 8
Solve −2x2 + 5x + 12 ≥ 0
Solution
Multiply the entire expression by -1 and change the inequality sign
−2x2 + 5x + 12 ≥ 0 ⟹2x2 − 5x − 12 ≤ 0
Factorize the expression to get;
(2x + 3) (x − 4) ≤ 0.
Solve the roots;
(2x + 3) ≤ 0 ⟹ x ≤ -3/2.
(x − 4) ≤ 0 ⟹ x ≤ 4.
By applying the rule; (x – a) (x – b) ≥ 0, then a ≤ x ≤ b, we can comfortably write the
solutions of this quadratic inequality as:
-3/2 ≤ x ≤ 4.