Grade by Grade
C. Polynomial and Rational Functions · Rational Functions and Equations

Rational Inequalities: Equations versus Inequalities, and Graphical Solutions

C4.1 C4.2About 70 minutes
Learning goals
  • I can explain the difference between the solution of a rational equation and the solution of a rational inequality.
  • I can show that given values do or do not satisfy an inequality by substituting them.
  • I can solve rational inequalities from graphs, including f(x) > k and f(x) > g(x).
  • I can write solutions in interval notation, in set notation and on a number line, with the right endpoints included or excluded.

Words to know

inequality
A statement that compares two expressions with <, >, ≤ or ≥. Its solution is usually a set of intervals.
interval notation
A way to write sets of numbers: (a, b) excludes both ends, [a, b] includes both, and ∞ always gets a round bracket.
union (∪)
'Or': the set of numbers in either interval. (−∞, −1) ∪ [3, ∞) means x < −1 or x ≥ 3.
boundary value
An x-value where the solution of an inequality can start or stop: a zero, an intersection point, or a vertical asymptote.

When 'equal' is not the question

A pharmacist wants a medicine's concentration to stay above a certain level. A business wants its average cost below a target. These are inequalities. In the unit Polynomial Functions and Equations you solve polynomial inequalities; here you solve rational ones. The big new idea is that a rational function can change sign not only at its zeros but also at its vertical asymptotes.

Equation or inequality?

Compare

  • the equation (x + 3)/(x − 2) = 0, whose solution is the single value x = −3, and
  • the inequality (x + 3)/(x − 2) ≤ 0, whose solution is the whole interval −3 ≤ x < 2.

An equation asks, 'Where is the graph exactly at this height?' Its solution is usually a list of values. An inequality asks, 'Where is the graph above (or below) this height?' Its solution is usually one or more intervals.

To demonstrate that a value satisfies an inequality, substitute it:

  • x = 0: 3/(−2) = −1.5 ≤ 0 ✓ (a solution);
  • x = −3: 0/(−5) = 0 ≤ 0 ✓ (a solution, because of the 'or equal to');
  • x = 2: undefined, so not a solution;
  • x = 5: 8/3 > 0, so not a solution.
y = (x + 3)/(x − 2): where is y ≤ 0?xy−7−6−5−4−2−1134567−5−4−3−2−123x = 2y = 1(−3, 0)graph on or below the x-axis
The shaded strip −3 ≤ x < 2 is where the graph is on or below the x-axis. The point (−3, 0) is included because the inequality allows 0. x = 2 is excluded because the function is undefined there.

Reading the solution from a graph

  • f(x) > 0: where the graph is above the x-axis. f(x) < 0: where it is below.
  • f(x) > k: where the graph is above the horizontal line y = k.
  • f(x) > g(x): where the graph of f is above the graph of g.

The solution can only start or stop at boundary values: zeros, intersection points and vertical asymptotes.

  • For ≤ or ≥, include zeros and intersection points (closed circle, square bracket).
  • For < or >, exclude them (open circle, round bracket).
  • Never include a vertical asymptote: the function is undefined there.
Solution of (x + 3)/(x − 2) ≤ 0: −3 ≤ x < 2sign+−+−3(zero)2(undefined)
The same solution on a number line. The closed circle at −3 means 'included'; the open circle at 2 means 'excluded'. In interval notation: [−3, 2).
Worked example

Example 1: f(x) > k from a graph

Solve 1/(x − 2) > 3 using a graph.

  1. Graph y = 1/(x − 2) and the line y = 3.
  2. Find the intersection: 1/(x − 2) = 3 gives x − 2 = 1/3, so x = 7/3.
  3. The graph is above y = 3 only on the right branch, between the asymptote x = 2 and the intersection x = 7/3. (The left branch is negative, so it is never above 3.)
  4. The inequality is strict (>), so exclude 7/3; always exclude the asymptote 2.

2 < x < 7/3, or (2, 7/3).

Worked example

Example 2: Including an endpoint

Solve x/(x − 1) ≥ 2 using a graph.

  1. x/(x − 1) = 1 + 1/(x − 1): asymptotes x = 1 and y = 1, decreasing on each branch.
  2. Intersection with y = 2: x = 2(x − 1), so x = 2.
  3. The left branch (x < 1) stays below y = 1, so it is never ≥ 2.
  4. On the right branch the graph comes down from +∞ and reaches 2 at x = 2, then keeps falling toward 1.
  5. '≥' includes x = 2.

1 < x ≤ 2, or (1, 2].

1/(x − 2) > 3xy12345−3−2−1123456x = 2y = 3(7/3, 3)x/(x − 1) ≥ 2xy−2−11234−3−2−1123456x = 1y = 1y = 2(2, 2)
Left: y = 1/(x − 2) is above y = 3 only for 2 < x < 7/3. Right: y = x/(x − 1) is on or above y = 2 for 1 < x ≤ 2. The shaded strips are the solutions.
Worked example

Example 3: Sign from a familiar graph

Solve 1/(x² − 4) < 0.

  1. From the lesson on reciprocals of quadratics: VAs x = ±2, and the middle branch is below the x-axis while the outer branches are above.
  2. So the graph is below the x-axis exactly on the middle branch.

−2 < x < 2, or (−2, 2), or {x ∈ ℝ | −2 < x < 2}.

Solving with technology

To solve f(x) > g(x) with a graphing tool, either graph both and look for where f is above g, or graph y = f(x) − g(x) and look for where it is above the x-axis. Use the intersect or zero command to get the boundary values exactly or to several decimal places. Then decide, using the inequality sign, whether each boundary is included. Technology does not decide that for you, and it may draw an asymptote as a near-vertical line, so always check boundaries by hand.

Just for fun

Equations want one exact answer. Inequalities are happy with a whole neighbourhood of answers, as long as nobody tries to move in right on the asymptote.

Hands-on activity

Human sign chart

You need: masking tape (or a long piece of string), about 15 sticky notes or scraps of paper, a marker, and enough floor space to walk about 3 m.

  1. Tape a long number line on the floor. Mark −5 to 5 with sticky notes, about 30 cm apart.
  2. Choose the inequality (x + 3)/(x − 2) ≤ 0. Put a sticky note with a filled dot at −3 (a zero) and one with a dashed line at 2 (an asymptote).
  3. Stand on different numbers: −5, −3, −1, 0, 1.9, 2, 3 and 5. At each one, calculate the value of (x + 3)/(x − 2) aloud (use a calculator). Say 'yes' if it is ≤ 0 and step forward, 'no' if not and step back.
  4. Walk the 'yes' region from end to end. Where does it start? Where does it stop? Is each end included?

What to look for: values 0.29, 0, −0.67, −1.5, −49, undefined, 6 and 2.67. The 'yes' region runs from −3 (included, value 0) up to but not including 2 (undefined): −3 ≤ x < 2. Everyone who stood at 2 learned that an asymptote is never part of the answer.

Your turn

1

Explain the difference between the solutions of 2/(x + 1) = 1 and 2/(x + 1) > 1.

2

Show that x = −2 satisfies (x − 1)/(x + 3) < 0.

3

Is x = −3 a solution of (x − 1)/(x + 3) ≤ 0? Explain.

4

Solve (x − 1)/(x + 3) < 0 using the graph or a sign chart.

5

Use the graph of y = 1/(x² − 4) to solve 1/(x² − 4) > 0.

6

Solve 1/(x + 1) ≥ 1 graphically.

7

Write −3 ≤ x < 2 in interval notation and in set notation.

More practice: warm-up

1

Write x > 4 in interval notation.

2

Is x = 0 a solution of 1/(x − 1) < 0?

3

Solve the equation (x − 5)/(x + 1) = 0.

4

Solve 1/x > 0.

5

Can the x-value of a vertical asymptote ever be part of the solution of a rational inequality?

More practice: on your own

1

Solve (2x − 4)/(x + 1) ≥ 0.

2

Solve 2/(x − 3) < −1 graphically.

3

Solve x/(x + 2) ≤ 3 graphically.

4

The solution of 1/(x − 2) < 3 is x < 2 or x > 7/3. Show that x = 2.5 and x = 0 satisfy it and x = 2.2 does not.

5

Solve 1/(x² + 1) > 1/2 graphically.

Challenge

1

Solve 1/(x² − 4) ≤ −1/4.

2

Solve (x + 1)/(x − 1) > x. (Hint: find where the graphs meet, then decide which is on top.)

3

For which k does 1/(x² + 1) > k have (a) no solution (b) every real number as its solution?

Check your understanding

Quiz

1.

Which statement best describes the difference between solving a rational equation and solving a rational inequality?

2.

What is the solution of (x + 3)/(x − 2) ≤ 0?

3.

Which value is a solution of 1/(x − 2) > 3?

4.

What is the solution of 1/(x² − 4) < 0?

5.

On a graph, where is f(x) > g(x)?

6.

How is 'x < −1 or x ≥ 3' written in interval notation?

For parents and teachers

Big idea (C4.1, C4.2): an equation's solution is usually a few values (where the graph hits a height); an inequality's solution is usually one or more intervals (where the graph is above or below). For rational functions, the solution can start or stop at a vertical asymptote, but never includes it.

How to help: Ask your teen to test one value from inside their answer and one from outside it. If the inside value works and the outside value fails, the answer is probably right.

Common difficulties:

  • Including a vertical asymptote value in the answer (for example writing −3 ≤ x ≤ 2).
  • Forgetting to include an endpoint when the sign is ≤ or ≥.
  • Mixing up round and square brackets, and using ∩ (and) instead of ∪ (or).
  • Reading the y-values instead of the x-values off the graph.