# Rational Inequality

How to solve a rational inequality: 2 examples and their solutions.

## Example 1

### Example

### Solution

Before solving the inequality,

first set (denominator) ≠ 0.

Excluded Value

See 1/x.

The denominator is x.

So x ≠ 0.

Next, see 1/2x.

The denominator is 2x.

So 2x ≠ 0.

Then x ≠ 0.

From the denominators,

you found that

x ≠ 0.

The x values should satisfy this condition.

Next, solve the rational inequality.

First, find the least common multiple, LCM,

of the denominators.

The denominators are

x and 2x.

So the LCM is

2x.

Change the denominators of the rational expression

to the LCM: 2x.

Add and Subtract Rational Expressions

See 1/x.

The denominator is x.

The factor 2 is missing.

So multiply 2

to both of the numerator and the denominator.

1/x

= [1/x]⋅[2/2]

See -1/2x.

The denominator is 2x:

the LCM.

So you don't have to change -1/2x.

Write the inequality sign ≥.

See the right side 3.

The denominator is 1.

2x is missing.

So multiply 2x

to both of the numerator and the denominator.

3

= 3⋅[2x/2x]

So

1/x - 1/2x ≥ 3

becomes

[1/x]⋅[2/2] - 1/2x ≥ 3⋅[2x/2x].

[1/x]⋅[2/2]

= 2/2x

3⋅[2x/2x] = 6x/2x

2/2x - 1/2x = 1/2x

Move 6x/2x to the right side.

Multiply -1 to both sides.

Then (6x - 1)/2x ≤ 0.

Multiplying (-) on both sides

changes the order of the inequality sign:

≥ → ≤.

Linear Inequality (One Variable)

Divide both sides by 2.

Then (6x - 1)/x ≤ 0.

Move the denominator x

to the numerator.

(6x - 1)/x → x(6x - 1)

This is

multiplying the square of the denominator, x^{2},

to both sides.

x ≠ 0

So you can multiply x^{2} on both sides.

And x^{2} is (+).

So the order of the inequality sign

doesn't change.

And write ≠ 0

under x.

Find the zeros of x(6x - 1).

Quadratic Equation: by Factoring

Case 1) x = 0

(We know that x ≠ 0.

This is the zero you're going to use

when graphing the inequality on the x-axis.)

Case 2) 6x - 1 = 0

Then x = 1/6.

Case 1) x = 0

Case 2) x = 1/6

So the zeros are x = 0, 1/6.

Draw y = x(6x - 1)

on the x-axis.

First point the zeros x = 0 and 1/6.

And draw a parabola

that passes through x = 0 and 1/6.

Quadratic Function: Find Zeros

x ≠ 0

So draw an empty circle on x = 0.

See x(6x - 1) ≤ 0.

The left side is less than or equal to 0.

So color the region

where the graph is below the x-axis (y = 0).

The inequality sign includes equal to [=].

So draw a full circle on x = 1/6.

(Don't fill the empty circle on x = 0.

This is the excluded value.)

The colored region is

0 < x ≤ 1/6.

So

0 < x ≤ 1/6

is the answer.

## Example 2

### Example

### Solution

Before solving the inequality,

first set (denominator) ≠ 0.

See 4/(x - 1).

The denominator is (x - 1).

So x - 1 ≠ 0.

Then x ≠ 1.

Next, see 1/x.

The denominator is x.

So x ≠ 0.

From the denominators,

you found that

x - 1 ≠ 0, x ≠ 0.

The x values should satisfy these conditions.

Next, solve the rational inequality.

First, find the least common multiple, LCM,

of the denominators.

The denominators are

(x - 1) and x.

So the LCM is

x(x - 1).

Change the denominators of the rational expression

to the LCM: x(x - 1).

See 4/(x - 1).

The denominator is (x - 1).

The factor x is missing.

So multiply x

to both of the numerator and the denominator.

4/(x - 1)

= [4/(x - 1)]⋅[x/x]

See the next term +1.

The denominator is 1.

x(x - 1) is missing.

So multiply x(x - 1)

to both of the numerator and the denominator.

+1

= +1⋅[[x(x - 1)]/[x(x - 1)]]

Write the inequality sign ≤.

See the right side 1/x.

The denominator is x.

The factor (x - 1) is missing.

So multiply (x - 1)

to both of the numerator and the denominator.

1/x

= [1/x]⋅[(x - 1)/(x - 1)]

So

4/(x - 1) + 1 ≤ 1/x

becomes

[4/(x - 1)]⋅[x/x] + 1⋅[[x(x - 1)]/[[x(x - 1)]] ≤ [1/x]⋅[(x - 1)/(x - 1)].

4⋅x = 4x

+1⋅x(x - 1) = +x^{2} - x

Multiply a Monomial and a Polynomial

1⋅(x - 1) = x - 1

4x + x^{2} - x

= x^{2} + 3x

Move (x - 1)/[x(x - 1)] to the left side.

Then (x^{2} + 3x - x + 1)/[x(x - 1)].

+3x - x = +2x

x^{2} + 2x + 1

= x^{2} + 2⋅x⋅1 + 1^{2}

= (x + 1)^{2}

Factor a Perfect Square Trinomial

Move the denominator x(x - 1)

to the numerator.

(x + 1)^{2}/[x(x - 1)] → (x + 1)^{2}x(x - 1)

This is

multiplying the square of the denominator,

[x(x - 1)]^{2},

to both sides.

(x - 1) ≠ 0, x ≠ 0

So x(x - 1) ≠ 0.

So you can multiply [x(x - 1)]^{2} on both sides.

And [x(x - 1)]^{2} is (+).

So the order of the inequality sign

doesn't change.

And write ≠ 0

under x and (x - 1).

Write the zeros.

x = -1, 0, 1

(We know that x ≠ 0, 1.

These are the zeros you're going to use

when graphing the inequality on the x-axis.)

Draw the x-axis.

Point the zeros x = -1, 0, 1.

Draw y = (x + 1)^{2}x(x - 1)

on the x-axis.

Polynomial Inequality

The highest degree term of y = (x + 1)^{2}x(x - 1)

is x^{4}.

The coefficient is (+).

So starting from the top right of the x-axis,

draw the graph

that goes toward the nearest zero:

x = 1.

See the factor (x - 1).

(x - 1) = (x - 1)^{1}

The exponent is 1.

It's odd.

Then draw the graph

that passes through the x-axis

at x = 1.

Next, see the factor x.

x = x^{1}

The exponent is 1.

It's odd.

Then draw the graph

that passes through the x-axis

at x = 0.

See the factor (x + 1)^{2}.

The exponent is 2.

It's even.

Then draw the graph

that bounces off the x-axis

at x = -1.

So this is the graph of the polynomial

y = (x + 1)^{2}x(x - 1)

on the x-axis.

x ≠ 0, (x - 1) ≠ 0

So draw empty circles on x = 0 and x = 1.

See (x + 1)^{2}x(x - 1) ≤ 0.

The left side is less than 0.

So color the region

where the graph is below the x-axis (y = 0).

The inequality sign does not include equal to [=].

So draw a full circle on x = -1.

(Don't fill the empty circles on x = 0 and x = 1.

These are the excluded values.)

The colored regions are

x = -1, 0 < x < 1.

So

x = -1, 0 < x < 1

is the answer.