Wize High School Algebra II Textbook (Common Core) > Rational Functions

Solving Rational Inequalities (Number Line & Graphs)

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Solving Rational Inequalities 
Rational inequalities can be solved in a similar way to rational expressions. 

Step 1. 
Identify any non-permissible values.

Step 2. 
Combine all terms to the left side of the inequality and let the right-hand side equal 0. 

Step 3. 
Simplify the rational inequality, factoring when necessary. 

Step 4. 
Use a graph or a number line to determine where the rational expression is positive and negative. 

Step 5. 
State the intervals where the rational expression is positive or negative, identify any extraneous solutions. 

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Example

Solve for xxx using both a number line and a graph.

x−8x≤3−x\displaystyle\frac{x-8}{x}\leq{}3-xxx−8​≤3−x

Step 1. 
x≠0x\neq{}0x=0

Step 2. 

x−8x≤3−xx−8x−(3−x)≤0\begin{array}{rcl}
\displaystyle\frac{x-8}{x}&\leq{}&3-x\\\\
\displaystyle\frac{x-8}{x}-(3-x)&\leq&0\\\\
\end{array}xx−8​xx−8​−(3−x)​≤≤​3−x0​

Step 3. 

x−8x−(3−x)≤0(x−8)−x(3−x)x≤0x−8−3x+x2x≤0x2−2x−8x≤0(x−4)(x+2)x≤0\begin{array}{rcl}
\displaystyle\frac{x-8}{x}-(3-x)&\leq&0\\\\
\displaystyle\frac{(x-8)-x(3-x)}{x}&\leq{}&0\\\\
\displaystyle\frac{x-8-3x+x^2}{x}&\leq{}&0\\\\
\displaystyle\frac{x^2-2x-8}{x}\leq{}&0\\\\
\displaystyle\frac{(x-4)(x+2)}{x}&\leq{}&0
\end{array}xx−8​−(3−x)x(x−8)−x(3−x)​xx−8−3x+x2​xx2−2x−8​≤x(x−4)(x+2)​​≤≤≤0≤​0000​

Step 4. 
Number Line: 
X-intercepts at x=−2,4x=-2, 4x=−2,4
Vertical Asymptotes: x=0x=0x=0
Test Points: -3, -1, 1, 5
−3−115x−−++x−4−−−+x+2−+++(x−4)(x+2)x−+−+\begin{array}{|c|c|c|c|c|}\hline
&-3&-1&1&5\\\hline\\
x&-&-&+&+\\\\\hline\\
x-4&-&-&-&+\\\\\hline\\
x+2&-&+&+&+\\\\\hline\\
\displaystyle\frac{(x-4)(x+2)}{x}&-&+&-&+\\\\\hline
\end{array}xx−4x+2x(x−4)(x+2)​​−3−−−−​−1−−++​1+−+−​5++++​​      

Graphing:
X-intercepts at x=−2,4x=-2, 4x=−2,4
Vertical Asymptotes: x=0x=0x=0
Horizontal Oblique Asymptotes: Obliqe asymptote at y=x−2y=x-2y=x−2
Test Points: -3, -1, 1, 5 
−3−115x−−++x−4−−−+x+2−+++(x−4)(x+2)x−+−+\begin{array}{|c|c|c|c|c|}\hline
&-3&-1&1&5\\\hline\\
x&-&-&+&+\\\\\hline\\
x-4&-&-&-&+\\\\\hline\\
x+2&-&+&+&+\\\\\hline\\
\displaystyle\frac{(x-4)(x+2)}{x}&-&+&-&+\\\\\hline
\end{array}xx−4x+2x(x−4)(x+2)​​−3−−−−​−1−−++​1+−+−​5++++​​      


Step 5. 
We are looking for the part of the rational expression that lies below y=0.y=0.y=0.

(−∞,2] ∪ (0,4](-\infin,2]~\cup~(0,4](−∞,2] ∪ (0,4]

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Example: Solving Rational Inequalities 
Solve for x. 

x−4x+1≤x+3x−2\displaystyle\frac{x-4}{x+1}\leq{}\displaystyle\frac{x+3}{x-2}x+1x−4​≤x−2x+3​

Step 1.

The NPV's, which are also equivalent to the vertical asymptotes, are x≠−1,2x\neq-1,2x=−1,2.


Step 2.

x−4x+1≤x+3x−2x−4x+1−x+3x−2≤0\begin{array}{rcl}
\displaystyle\frac{x-4}{x+1}&\leq{}&\displaystyle\frac{x+3}{x-2}\\\\
\displaystyle\frac{x-4}{x+1}-\displaystyle\frac{x+3}{x-2}&\leq{}&0
\end{array}x+1x−4​x+1x−4​−x−2x+3​​≤≤​x−2x+3​0​


Step 3.

x−4x+1−x+3x−2≤0(x−4)(x−2)−(x+3)(x+1)(x+1)(x−2)≤0(x2−6x+8)−(x2+4x+3)(x+1)(x−2)≤0−10x+5(x+1)(x−2)≤0−5(2x−1)(x+1)(x−2)≤0\begin{array}{rcl}
\displaystyle\frac{x-4}{x+1}-\displaystyle\frac{x+3}{x-2}&\leq{}&0\\\\
\displaystyle\frac{(x-4)(x-2)-(x+3)(x+1)}{(x+1)(x-2)}&\leq{}&0\\\\
\displaystyle\frac{(x^2-6x+8)-(x^2+4x+3)}{(x+1)(x-2)}&\leq{}&0\\\\
\displaystyle\frac{-10x+5}{(x+1)(x-2)}&\leq{}&0\\\\
\displaystyle\frac{-5(2x-1)}{(x+1)(x-2)}&\leq{}&0
\end{array}x+1x−4​−x−2x+3​(x+1)(x−2)(x−4)(x−2)−(x+3)(x+1)​(x+1)(x−2)(x2−6x+8)−(x2+4x+3)​(x+1)(x−2)−10x+5​(x+1)(x−2)−5(2x−1)​​≤≤≤≤≤​00000​


Step 4.

X-Intercepts: 12\displaystyle\frac{1}{2}21​
Vertical Asymptotes: x=−1,2x=-1,2x=−1,2
Horizontal Asymptotes: y=0y=0y=0

Test Points: -2, 0, 1, 3

−2013−5−−−−(2x−1)−−++(x+1)−+++(x−2)−−−+−5(2x−1)(x+1)(x−2)+−+−\begin{array}{|c|c|c|c|c|}\hline
&-2&0&1&3\\\hline\\
-5&-&-&-&-\\\\\hline\\
(2x-1)&-&-&+&+\\\\\hline\\
(x+1)&-&+&+&+\\\\\hline\\
(x-2)&-&-&-&+\\\\\hline\\
\displaystyle\frac{-5(2x-1)}{(x+1)(x-2)}&+&-&+&-\\\\\hline
\end{array}−5(2x−1)(x+1)(x−2)(x+1)(x−2)−5(2x−1)​​−2−−−−+​0−−+−−​1−++−+​3−+++−​​      



Step 5.

We want the part of the rational expression that is less than or equals to 0. 

(−1,12] ∪ (2,∞)\Bigg(-1,\displaystyle\frac{1}{2}\Bigg]~\cup~(2,\infin)(−1,21​] ∪ (2,∞)

Practice: Solving Rational Inequalities 
The following is a graph of f(x)f(x)f(x).


Solve for the following polynomial inequalities. Express the solution in interval notation.

fx)<0

f(x)\geq{}0

when

x<0

f(x)>0

when

x>0

I don't know

Practice: Solving Rational Inequalities 

Find the values of x that make 2x−3x+2−3<2x\displaystyle\frac{2x-3}{x+2}-3<\displaystyle\frac{2}{x}x+22x−3​−3<x2​ true. 

A) (−∞,−11−1052] ∪ (−2,−11+1052) ∪ (0,∞)\Bigg(-\infin,\displaystyle\frac{-11-\sqrt{105}}{2}\Bigg]~\cup~\Bigg(-2,\displaystyle\frac{-11+\sqrt{105}}{2}\Bigg)~\cup~(0,\infin)(−∞,2−11−105​​] ∪ (−2,2−11+105​​) ∪ (0,∞) 

B) (−∞,−11−1052) ∪ (−2,−11+1052) ∪ (0,∞)\Bigg(-\infin,\displaystyle\frac{-11-\sqrt{105}}{2}\Bigg)~\cup~\Bigg(-2,\displaystyle\frac{-11+\sqrt{105}}{2}\Bigg)~\cup~(0,\infin)(−∞,2−11−105​​) ∪ (−2,2−11+105​​) ∪ (0,∞)  

C) (−∞,−11−1052] ∪ [−2,−11+1052] ∪ [0,∞)\Bigg(-\infin,\displaystyle\frac{-11-\sqrt{105}}{2}\Bigg]~\cup~\Bigg[-2,\displaystyle\frac{-11+\sqrt{105}}{2}\Bigg]~\cup~[0,\infin)(−∞,2−11−105​​] ∪ [−2,2−11+105​​] ∪ [0,∞) 

D) None of the above 

I don't know

Practice: Solving Rational Inequalities 
Solve for x. 

∣2x+2∣≥4\Bigg|\displaystyle\frac{2}{x+2}\Bigg|\geq{}4​x+22​​≥4

A) (−52,−2] ∪ [−2,32)\Bigg(-\displaystyle\frac{5}{2},-2\Bigg]~\cup~\Bigg[-2,\displaystyle\frac{3}{2}\Bigg)(−25​,−2] ∪ [−2,23​) 

B) [−52,−2] ∪ [−2,32]\Bigg[-\displaystyle\frac{5}{2},-2\Bigg]~\cup~\Bigg[-2,\displaystyle\frac{3}{2}\Bigg][−25​,−2] ∪ [−2,23​] 

C) (−52,−2) ∪ (−2,32)\Bigg(-\displaystyle\frac{5}{2},-2\Bigg)~\cup~\Bigg(-2,\displaystyle\frac{3}{2}\Bigg)(−25​,−2) ∪ (−2,23​) 

D) [−52,−2) ∪ (−2,32]\Bigg[-\displaystyle\frac{5}{2},-2\Bigg)~\cup~\Bigg(-2,\displaystyle\frac{3}{2}\Bigg][−25​,−2) ∪ (−2,23​] 

I don't know

Extra Practice

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
Solve for x. 
∣3x∣>x\Bigg|\dfrac{3}{x}\Bigg|>x​x3​​>x

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
Solve for x. 
∣1x−3∣≥1\Bigg|\dfrac{1}{x-3}\Bigg|\geq{}1​x−31​​≥1

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
Find the values of x that make x−2x+4−1<1x\dfrac{x-2}{x+4}-1<\dfrac{1}{x}x+4x−2​−1<x1​ true. 

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
Find the values of x that make 5x+9x−1+3>−3x\dfrac{5x+9}{x-1}+3>-\dfrac{3}{x}x−15x+9​+3>−x3​ true. 

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
The following is a graph of f(x)f(x)f(x).
Solve for the following polynomial inequalities. Express the solution in interval notation.

Solving Rational Inequalities

Practice: Solving Rational Inequalities 
The following is a graph of f(x)f(x)f(x).
Solve for the following polynomial inequalities. Express the solution in interval notation.

Solving Polynomial Inequalities

Solve for bbb.
5b+67=b+3b+1\dfrac{5b+6}{7}=\dfrac{b+3}{b+1}75b+6​=b+1b+3​

Rational Inequalities

Determine an expression for a polynomial, f(x),f(x),f(x), that has the following properties:
f(x)≤0f(x)\leq{}0f(x)≤0 when x≤4x\leq{}4x≤4
f(x)≥0f(x)\geq{}0f(x)≥0 when x≥4x\geq{}4x≥4

Solving Rational Inequalities

Determine x:x:x:  3xx2+5x+6<2x+1\dfrac{3x}{x^2+5x+6}<\dfrac{2}{x+1}x2+5x+63x​<x+12​

Solving Rational Inequalities (Number Line & Graphs)

Solve for a:a:a: 
a−3a2+2a+1≥5a+1\dfrac{a-3}{a^2+2a+1}\geq\dfrac{5}{a+1}a2+2a+1a−3​≥a+15​

Inequalities

Solve 3x+5≥x2+13x+5\geq{}x^2+13x+5≥x2+1.