Linear and Absolute Value Inequalities: Concepts, Properties, and Applications

Linear Inequalities in One Variable

Definition and Forms

Linear inequalities are mathematical statements involving a linear expression in one variable that use inequality symbols (<, >, ≤, ≥) instead of an equality. The general form is ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c, where a, b, and c are real numbers and a ≠ 0.

• Solution Set: The set of all values of x that satisfy the inequality. Solutions can be expressed in interval notation or set-builder notation.

• Graphical Representation: Solutions are often shown on a number line, with open or closed circles indicating whether endpoints are included.

Interval Notation and Graphs

Interval notation is a concise way to describe sets of numbers between endpoints. Brackets [ ] indicate inclusion of endpoints; parentheses ( ) indicate exclusion.

Inequality	Interval Notation
a ≤ x ≤ b	[a, b]
a < x < b	(a, b)
a ≤ x < b	[a, b)
x ≥ a	[a, ∞)
x > a	(a, ∞)
x ≤ b	(−∞, b]
x < b	(−∞, b)

Note: [ ] means the endpoint is included; ( ) means it is not.

Properties of Inequality

Let a, b, and c be real numbers:

• If a < b, then a + c < b + c.

• If a < b and b < c, then a < c.

• If a < b and c > 0, then ac < bc.

• If a < b and c < 0, then ac > bc.

Solving Linear Inequalities

Steps to Solve

Solving a linear inequality is similar to solving a linear equation, with attention to the direction of the inequality when multiplying or dividing by a negative number.

1. Isolate the variable on one side.

2. Simplify both sides as needed.

3. If you multiply or divide by a negative, reverse the inequality sign.

4. Express the solution in interval notation and graph it if required.

Examples

• Example 1: Solve $-8x + 1 < 17$

• Example 2: Solve $8 = -\frac{5}{2}x - 2$

• Example 3: Solve $-14 < 3(x - 7) + 7$

• Example 4: Solve $5(7 - x) + 2x + 6x - 2 - 9x$

• Example 5: Solve $4 - 3x \geq -2(x + 3) - x$

Compound Inequalities

Definition and Solution Methods

A compound inequality consists of two inequalities joined by "and" or "or". The solution set depends on the logical connector:

• "And": Solutions must satisfy both inequalities (intersection).

• "Or": Solutions satisfy at least one inequality (union).

Steps to Solve Compound Inequalities

1. Solve each inequality separately.

2. Combine the solution sets using intersection (and) or union (or).

3. Express the final solution in interval notation and graph it.

Examples

• Example 1: Solve $x \leq -2$ and $x > -5$

• Example 2: Solve $x - 6 > 1$ and $\frac{1}{2}x \geq 6$

• Example 3: Solve $-\frac{4}{5}x < 8$ and $0.85 \leq 0.34x$

• Example 4: Solve $3(x - 2) + 2 \leq x - 8$ and $4(x + 1) + 2 > -2x + 4$

Absolute Value Inequalities

Definition and Properties

The absolute value of a real number x, denoted |x|, is its distance from zero on the number line. Absolute value inequalities involve expressions like |x| < k or |x| > k, where k > 0.

• |x| < k is equivalent to $-k < x < k$

• |x| > k is equivalent to $x < -k$ or $x > k$

These properties also apply to ≤ and ≥.

Graphical Representation

Inequality	Graph Description	Solution Set
|x| < 3	Values between -3 and 3	$-3 < x < 3$
|x| > 3	Values less than -3 or greater than 3	$x < -3$ or $x > 3$

Examples

• Example 1: Solve $|x| = 8$

• Example 2: Solve $|x| < 8$

• Example 3: Solve $|x + 1| - 4 \leq 1$

• Example 4: Solve $|2x + 7| + 5 < 1$

• Example 5: Solve $-18 \leq 6 - |3x + 3|$

Applications of Absolute Value Inequalities

Measurement Error and Real-World Contexts

Absolute value inequalities are used to express tolerances and errors in measurements. For example, if a board is to be cut to 24 inches with a maximum error of 0.02 inches, the actual length x must satisfy:

• $|x - 24| \leq 0.02$

This describes all values of x within 0.02 units of 24.

Example: Snowfall Data

Given monthly snowfall data for Chicago, Illinois, absolute value inequalities can be used to determine how much snowfall is needed to exceed an average value.

• Given monthly values: 2.8, 8.4, 11.2, 7.9 inches.

• Find the value for March so that the average exceeds 7.28 inches.

Additional info: This type of problem requires setting up an equation for the average and solving for the unknown month.

Summary Table: Properties of Absolute Value Inequalities

Form	Equivalent Statement
|x| < k	$-k < x < k$
|x| > k	$x < -k$ or $x > k$
|x| \leq k	$-k \leq x \leq k$
|x| \geq k	$x \leq -k$ or $x \geq k$

Key Concepts and Takeaways

• Linear inequalities describe ranges of values for variables and are solved using algebraic manipulation.

• Compound inequalities require careful consideration of logical connectors (and/or).

• Absolute value inequalities express distances from a central value and are widely used in measurement and error analysis.

• Always reverse the inequality sign when multiplying or dividing by a negative number.