2.7 Solve Absolute Value Inequalities

Absolute value equations and inequalities are powerful tools in algebra. They help us understand distances, ranges, and tolerances in real-world situations. By mastering these concepts, you'll be able to solve complex problems and apply them to practical scenarios.

Solving absolute value equations involves isolating the absolute value expression and considering both positive and negative solutions. For inequalities, we split them into two parts and use compound statements to represent the solution set graphically on a number line.

Solving Absolute Value Equations and Inequalities

Equations with absolute value

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• Isolate the absolute value expression on one side of the equation by performing inverse operations (addition, subtraction, multiplication, or division) on both sides
• If the equation is in the form $[|x|](https://www.fiveableKeyTerm:|x|) = a$, where $a \geq 0$, then the solutions are $x = a$ or $x = -a$
  • Absolute value always results in a non-negative value, so the equation $|x| = a$ has two solutions: the positive and negative values of $a$ ($5$ and $-5$)
• If the equation is in the form $|x| = a$, where $a [<](https://www.fiveableKeyTerm:<) 0$, then there is no solution because absolute value cannot be negative
  • Absolute value of a number is always non-negative, so an equation like $|x| = -3$ has no real solution
• Solving absolute value equations often requires algebraic manipulation to isolate the absolute value expression

Inequalities: absolute value less than

• For an inequality in the form $|x| < a$, where $a [>](https://www.fiveableKeyTerm:>) 0$, the solutions lie between $-a$ and $a$ (exclusive)
  • The inequality $|x| < 4$ has solutions in the range $-4 < x < 4$, meaning $x$ can be any value between $-4$ and $4$, but not including $-4$ or $4$
• Split the inequality into two parts: $-a < x$ and $x < a$
  • Combine the two inequalities using "and" since both conditions must be satisfied simultaneously (compound statements)
• Graph the solution on a number line using open circles to represent strict inequalities (< or >)
  • Open circles indicate that the endpoint values are not included in the solution set

Inequalities: absolute value greater than

• For an inequality in the form $|x| > a$, where $a \geq 0$, the solutions lie outside the range $-a$ to $a$ (exclusive)
  • The inequality $|x| > 3$ has solutions $x < -3$ or $x > 3$, meaning $x$ can be any value less than $-3$ or greater than $3$
• Split the inequality into two parts: $x < -a$ or $x > a$
  • Combine the two inequalities using "or" since either condition can be satisfied (logical connectors)
• Graph the solution on a number line using open circles to represent strict inequalities (< or >)
  • The solution set consists of two separate regions on the number line

Real-world applications of absolute value

• Distance: Absolute value represents the distance between two points on a number line, regardless of direction
  • If a town is located at mile marker 50 and a gas station is 15 miles away, the gas station could be at mile marker 35 ($50 - 15$) or 65 ($50 + 15$)
• Tolerance: Absolute value represents the maximum allowed deviation from a target value in manufacturing or quality control
  • If a machine part must have a length of 10 cm with a tolerance of 0.5 cm, the acceptable range is $9.5 \leq x \leq 10.5$, which can be written as $|x - 10| \leq 0.5$
• Range: Absolute value defines a range of values within a certain distance from a central point
  • If a store wants to target customers living within 5 miles of its location, the range can be expressed as $|x| \leq 5$, where $x$ is the distance from the store in miles

Set Theory and Inequality Properties

• Set theory is used to describe solution sets of absolute value inequalities
• Inequality properties are applied when solving absolute value inequalities:
  • Addition and subtraction property: adding or subtracting the same value from both sides of an inequality preserves the inequality
  • Multiplication and division property: multiplying or dividing both sides by a positive number preserves the inequality, while multiplying or dividing by a negative number reverses the inequality sign