Express the statement as an inequality. (a) b is positive....

Express the statement as an inequality. (a) $b$ is positive. (b) $s$ is nonpositive. (c) $w$ is greater than or equal to $-4$. (d) $c$ is between $\frac{1}{5}$ and $\frac{1}{3}$. (e) $p$ is not greater than $-2$. (f) The negative of $m$ is not less than $-2$. (g) The quotient of $r$ and $s$ is at least $\frac{1}{s}$. (h) The reciprocal of $f$ is at most 14. (i) The absolute value of $x$ is less than 4.

Express the statement as an inequality.
(a) $b$ is positive.
(b) $s$ is nonpositive.
(c) $w$ is greater than or equal to $-4$.
(d) $c$ is between $\frac{1}{5}$ and $\frac{1}{3}$.
(e) $p$ is not greater than $-2$.
(f) The negative of $m$ is not less than $-2$.
(g) The quotient of $r$ and $s$ is at least $\frac{1}{s}$.
(h) The reciprocal of $f$ is at most 14.
(i) The absolute value of $x$ is less than 4.

Key Concepts

Reciprocal Relationships in Inequalities

When statements involve the reciprocal or inversion of a quantity, careful handling is required because taking reciprocals can affect the direction of the inequality depending on the sign of the variable. This concept is crucial when expressing inequalities that involve fractions or reciprocal expressions.

Translating Verbal Statements to Mathematical Inequalities

This concept entails taking a description in words and converting it to a corresponding algebraic inequality. It involves identifying key phrases that indicate the type of inequality (such as 'positive', 'nonpositive', 'at least', 'less than', etc.) and correctly applying the appropriate inequality symbols.

Absolute Value Inequalities

Absolute value inequalities define a range for the magnitude of a variable irrespective of its sign. Understanding how to represent and solve inequalities involving absolute values is important, as it encapsulates conditions where a variable's distance from zero is bounded by a specific number.

Strict vs. Non-Strict Inequalities

Key to interpreting and expressing mathematical statements is distinguishing between strict inequalities (using > or <, which indicate a clear separation between numbers) and non-strict inequalities (using ? or ?, which allow for equality as well as a greater or lesser value).

Inequality Notation

This concept involves understanding and using the symbols such as >, <, ?, and ? to represent relationships between quantities. It is fundamental for translating written descriptions into mathematical statements that denote one quantity’s size relative to another.

Compound Inequalities

Compound inequalities are used when a variable must satisfy more than one condition simultaneously, as in being between two values. This concept helps in representing and understanding ranges where a variable is bounded below, above, or both.

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