2.5 Solution of Quadratic Inequalities in One Unknown

2.5 Solution of Quadratic Inequalities in One Unknown

Inequality In Mathematics an inequality is a relation which makes a un-equal comparison between two numbers or other mathematical expressions. We have learned how to use the inequality symbols $<$ (less than), $>$ (greater than), $\le$ (less than or equal to) and $\ge$ (greater than or equal to).

Here we learn how to solve quadratic inequalities in one unknown.

There are following four steps to solve quadratic inequality:
(i) Solve the corresponding equation. (ii) Determine intervals.
(iii) Test points in each interval. (iv) Conclusion.

Example 16

Solve the inequality $x^2 - 5x + 6 > 0$.

Solution $x^2 - 5x + 6 > 0$

Step I Solve the corresponding or associated equation.

Set the quadratic expression equal to zero ($0$) to find the roots.

$x^2 - 5x + 6 = 0$

$x^2 - 3x - 2x + 6 = 0$

$(x - 2)(x - 3) = 0$ [Factorize]

Now, $x - 2 = 0$ or $x - 3 = 0$

So, the solutions are $x = 2$ and $x = 3$.

Here, critical points $(2, 0)$ and $(3, 0)$.

Step II Determine intervals

The critical points divide the number line into three intervals.

Interval 1: $x < 2$
Interval 2: $2 < x < 3$
Interval 3: $x > 3$

[Number line showing intervals $x<2$, $2<x<3$, $x>3$]

Step III Test points in each interval.

Choose a test point in each interval and substitute it into the inequality.

For $x = 1$: $(1 - 2)(1 - 3) = 2 > 0$.
So, this interval satisfies the inequality.

For $x = 2.5$: $(2.5 - 2)(2.5 - 3) = -0.25 < 0$.
So, this interval does not satisfy the inequality.

For $x = 4$: $(4 - 2)(4 - 3) = 2 > 0$.
So, this interval satisfies the inequality.

Step IV Conclusion:

The solution of this inequality $x^2 - 5x + 6 > 0$
is $x \in (-\infty, 2) \cup (3, \infty)$

Example 17

Solve the inequality $2x^2 - 5x + 2 \le 0$.

Solution $2x^2 - 5x + 2 \le 0$

Step I Solve the corresponding or associated equation.

$2x^2 - 5x + 2 = 0$

$2x^2 - 4x - x + 2 = 0$

$2x(x - 2) - 1(x - 2) = 0$

$(2x - 1)(x - 2) = 0$

Now, $2x - 1 = 0$ or $x - 2 = 0$

So, the solutions are $x = \dfrac{1}{2}$ and $x = 2$

Here, critical points are $\left(\dfrac{1}{2}, 0\right)$ and $(2, 0)$

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Note
For solution, we use open interval $(a, b)$ when equation has $<$ or $>$ sign and closed interval $[a, b]$ when equation has $\le$ or $\ge$ sign.

Step II Determine Intervals

The critical points $\left(\dfrac{1}{2}, 0\right)$, $(2, 0)$ divides the number line into three intervals.

Interval 1: $x < \dfrac{1}{2}$
Interval 2: $\dfrac{1}{2} < x < 2$
Interval 3: $x > 2$

[Number line showing intervals $x<1/2$, $1/2 \le x \le 2$, $x>2$]

Step III Test points in each interval.

For $x = 0$: $(2(0) - 1)(0 - 2) = (-1)(-2) = 2 > 0$,
So, this interval does not satisfy the inequality.

For $x = 1$: $(2(1) - 1)(1 - 2) = (1)(-1) = -1 < 0$,
So, this interval satisfies the inequality.

For $x = 3$: $(2(3) - 1)(3 - 2) = (6 - 1)(1) = 5 > 0$,
So, this interval does not satisfy the inequality.

Step IV Conclusion:

The solution of the inequality is: $x \in \left[\dfrac{1}{2}, 2\right]$.