1.7 Inequalities

1.7.1 Learning Objectives

By the end of this section, you should be able to:

solve linear and compound inequalities;
solve polynomial and rational inequalities;
solve absolute-value inequalities;
express solution sets using interval notation; and
build and interpret inequality models.

1.7.2 Linear and Compound Inequalities

An inequality compares quantities using $<$, $>$, $\le$, or $\ge$. Its solution usually consists of intervals.

Adding or subtracting the same expression on both sides preserves an inequality. Multiplying or dividing by a positive number also preserves it. Multiplying or dividing by a negative number reverses its direction:

\[ a<b\quad\Longrightarrow\quad -a>-b. \]

Example 1.33 Linear Inequality \[ 5-3x\le17 \Longrightarrow -3x\le12 \Longrightarrow x\ge-4. \] The solution is $\boxed{[-4,\infty)}$.

Exercise 1.39 Solve $4x+7>19$.

A compound inequality such as $a<x\le b$ requires both comparisons to be true.

Example 1.34 Compound Inequality \[ -7<2x+1\le9 \Longrightarrow -4<x\le4. \] The solution is $\boxed{(-4,4]}$.

Exercise 1.40 Solve $-3\le2x+1<9$.

1.7.3 Polynomial and Rational Inequalities

For a polynomial or rational inequality:

Move all terms to one side and factor.
Find values where a factor is zero or the expression is undefined.
Divide the real line at these critical numbers.
Test the sign on each interval.
Include permitted zeros for $\le$ or $\ge$, but never include undefined values.

Example 1.35 Quadratic Inequality \[ x^2-x-6=(x-3)(x+2)<0. \] The product is negative between its zeros, so the solution is $\boxed{(-2,3)}$.

Exercise 1.41 Solve $x^2-9\ge0$.

Exercise 1.42 Solve $(x-1)(x+4)\le0$.

Do not clear a variable denominator unless its sign is known. Multiplication by a negative expression reverses the inequality.

Example 1.36 Rational Inequality \[ \frac{x+1}{x-2}\ge0. \] Sign testing at the critical numbers $-1$ and $2$ gives \[ \boxed{(-\infty,-1]\cup(2,\infty)}. \]

1.7.4 Absolute-Value Inequalities

For $c>0$,

\[ |A|<c\iff -c<A<c, \qquad |A|>c\iff A<-c\text{ or }A>c. \]

Example 1.37 Absolute-Value Inequality \[ |2x-3|<7 \Longrightarrow -7<2x-3<7 \Longrightarrow \boxed{-2<x<5}. \]

Exercise 1.43 Solve $|x+2|\le5$.

1.7.5 Conceptual Takeaways

Inequalities commonly have intervals of solutions.
Multiplying or dividing by a negative reverses the comparison.
Critical numbers create sign intervals.
Denominator zeros are excluded.
Absolute-value inequalities describe distances.

1.7.6 Skills You Should Be Able to Do

Solve and graph linear and compound inequalities.
Use interval notation.
Build sign diagrams for polynomial and rational inequalities.
Solve absolute-value inequalities.
Model contextual restrictions.

1.7.7 Practice Problems with Solutions

Solve $7-5x<22$.

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$-5x<15$, so $\boxed{x>-3}$.
Solve $-2\le3x-5<10$.

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$3\le3x<15$, so $\boxed{1\le x<5}$.
Solve $x^2+2x-15\le0$.

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\[ (x+5)(x-3)\le0, \] so $\boxed{[-5,3]}$.
Solve $2x^2-x-6>0$.

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\[ (2x+3)(x-2)>0, \] so $\boxed{(-\infty,-3/2)\cup(2,\infty)}$.
Solve $\frac{x-4}{x+2}\le0$.

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Sign testing at $-2$ and $4$ gives $\boxed{(-2,4]}$.
Solve $|3x+1|\ge8$.

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\[ 3x+1\le-8\quad\text{or}\quad3x+1\ge8, \] so $\boxed{x\le-3\text{ or }x\ge7/3}$.
A tutoring plan costs $\$25$ plus $\$8$ per session. Another costs $\$13$ per session. When is the first plan less expensive?

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\[ 25+8n<13n\Longrightarrow n>5. \] For whole-number sessions, the first plan is cheaper for $\boxed{6\text{ or more sessions}}$.