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Wize High School Grade 12 Pre-Calculus Textbook > Inequalities

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 xx using both a number line and a graph.

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

Step 1.
x≠0x\neq{}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}

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}

Step 4.
Number Line:
X-intercepts at x=−2,4x=-2, 4
Vertical Asymptotes: x=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}

Graphing:
X-intercepts at x=−2,4x=-2, 4
Vertical Asymptotes: x=0x=0
Horizontal Oblique Asymptotes: Obliqe asymptote at y=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}


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

(−∞,2] ∪ (0,4](-\infin,2]~\cup~(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}


Step 1.

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


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}


Step 4.

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



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)

Practice: Solving Rational Inequalities

The following is a graph of f(x)f(x).


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

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} true.

Practice: Solving Rational Inequalities

Solve for x.

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



Extra Practice
Solving Rational Inequalities
Practice: Solving Rational Inequalities
Solve for x.
∣3x∣>x\Bigg|\dfrac{3}{x}\Bigg|>x
Solving Rational Inequalities
Practice: Solving Rational Inequalities
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∣1x−3∣≥1\Bigg|\dfrac{1}{x-3}\Bigg|\geq{}1
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Find the values of x that make x−2x+4−1<1x\dfrac{x-2}{x+4}-1<\dfrac{1}{x} true.
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Find the values of x that make 5x+9x−1+3>−3x\dfrac{5x+9}{x-1}+3>-\dfrac{3}{x} true.
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Practice: Solving Rational Inequalities
The following is a graph of 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).
Solve for the following polynomial inequalities. Express the solution in interval notation.
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Solve for bb.
5b+67=b+3b+1\dfrac{5b+6}{7}=\dfrac{b+3}{b+1}
Rational Inequalities
Determine an expression for a polynomial, f(x),f(x), that has the following properties:
f(x)≤0f(x)\leq{}0 when x≤4x\leq{}4
f(x)≥0f(x)\geq{}0 when x≥4x\geq{}4
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Determine x:x: 3xx2+5x+6<2x+1\dfrac{3x}{x^2+5x+6}<\dfrac{2}{x+1}
Solving Rational Inequalities (Number Line & Graphs)
Solve for a:a:
a−3a2+2a+1≥5a+1\dfrac{a-3}{a^2+2a+1}\geq\dfrac{5}{a+1}
Inequalities
Solve 3x+5≥x2+13x+5\geq{}x^2+1.