2.5 The Number Line Method - Introduction to Calculus

The Number Line Method

Another method for solving inequalities uses the following basic logic:

\begin{aligned} (+)(+)=+&\qquad\dfrac{(+)}{(+)}=+\\[.65em] (-)(-)=+&\qquad\dfrac{(-)}{(-)}=+\\[.65em] (+)(-)=-&\qquad\dfrac{(+)}{(-)}=-\\[.65em] (-)(+)=-&\qquad\dfrac{(-)}{(+)}=- \end{aligned}

By manipulating expressions into factors that are multiplied and/or divided on one side of the inequality (with a zero appearing on the other side), we can simply consider the combinations of positive and negative factors to draw conclusions.

Choose a factored expression and comparison. The signs determine the intervals; the comparison determines whether zeros are included.

Expression Comparison

\displaystyle (x-1)(3x-10)>0

Factor	$\displaystyle (-\infty,\,1)$	$\displaystyle (1,\,\frac{10}{3})$	$\displaystyle (\frac{10}{3},\,\infty)$
$\displaystyle x-1$	−	+	+
$\displaystyle 3x-10$	−	−	+
Product	+	−	+

Solution set:

\displaystyle (-\infty,\,1)\cup(\frac{10}{3},\,\infty)

Test input x0

Value at x = 0: 10.000 · Included in the solution.

Switch between > and ≥. Only the numerator zeros can be added; a denominator zero is always excluded.

QUESTION

For

\displaystyle \dfrac{(x-3)(x+2)}{x+1}\ge0

, which boundary is always excluded?

Find all values of $\displaystyle x$ that satisfy $\displaystyle 3x^2-13x>-10$

Find all values of $\displaystyle x$ that satisfy $\displaystyle \displaystyle x-2\ge\dfrac4{x+1}$

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The Number Line Method

Another method for solving inequalities uses the following basic logic:

\begin{aligned} (+)(+)=+&\qquad\dfrac{(+)}{(+)}=+\\[.65em] (-)(-)=+&\qquad\dfrac{(-)}{(-)}=+\\[.65em] (+)(-)=-&\qquad\dfrac{(+)}{(-)}=-\\[.65em] (-)(+)=-&\qquad\dfrac{(-)}{(+)}=- \end{aligned}

Choose a factored expression and comparison. The signs determine the intervals; the comparison determines whether zeros are included.

Expression Comparison

\displaystyle (x-1)(3x-10)>0

Factor	$\displaystyle (-\infty,\,1)$	$\displaystyle (1,\,\frac{10}{3})$	$\displaystyle (\frac{10}{3},\,\infty)$
$\displaystyle x-1$	−	+	+
$\displaystyle 3x-10$	−	−	+
Product	+	−	+

Solution set:

\displaystyle (-\infty,\,1)\cup(\frac{10}{3},\,\infty)

Test input x0

Value at x = 0: 10.000 · Included in the solution.

Switch between > and ≥. Only the numerator zeros can be added; a denominator zero is always excluded.

QUESTION

For

\displaystyle \dfrac{(x-3)(x+2)}{x+1}\ge0

, which boundary is always excluded?

Find all values of $\displaystyle x$ that satisfy $\displaystyle 3x^2-13x>-10$

Find all values of $\displaystyle x$ that satisfy $\displaystyle \displaystyle x-2\ge\dfrac4{x+1}$

The Number Line Method

The Number Line Method

Build a sign chart

KNOWLEDGE QUIZ

Use signs to solve a polynomial inequality

Use signs to solve a rational inequality

The Number Line Method

The Number Line Method

Build a sign chart

KNOWLEDGE QUIZ

Use signs to solve a polynomial inequality

Use signs to solve a rational inequality