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Unit 1: Linear Relationships and Equations

Topic 4

Linear Inequalities (Including Compound and Absolute Value)

An inequality shows a relationship where two expressions are not necessarily equal. Instead, one side is greater or less than the other.
a<bmeans a is less than ba>bmeans a is greater than ba≤bmeans a is less than or equal to ba≥bmeans a is greater than or equal to b\begin{aligned} a < b &\quad \text{means } a \text{ is less than } b \\ a > b &\quad \text{means } a \text{ is greater than } b \\ a \leq b &\quad \text{means } a \text{ is less than or equal to } b \\ a \geq b &\quad \text{means } a \text{ is greater than or equal to } b \end{aligned}
Solving inequalities is very similar to solving equations — the same operations can be performed on both sides. However, there is one key rule to remember:
When you multiply or divide by a negative number, you must flip the inequality sign.\textbf{When you multiply or divide by a negative number, you must flip the inequality sign.}
Compound Inequalities
Sometimes two inequalities are joined by the words and or or.
  • “and” means both conditions must be true — the overlap of the solution sets.
  • “or” means either condition can be true — the combined solution set.
Example of an “and” compound inequality:
−2<x≤5-2 < x \leq 5
This means xx is greater than −2-2 and less than or equal to 55.
Absolute Value Equations and Inequalities
The absolute value of a number represents its distance from 0 on the number line. Distance is always positive.
∣x∣={x,x≥0−x,x<0|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}
  • If ∣x∣=a|x| = a, then x=ax = a or x=−ax = -a.
  • If ∣x∣<a|x| < a, then −a<x<a-a < x < a.
  • If ∣x∣>a|x| > a, then x<−ax < -a or x>ax > a.

Core Skills

  • Apply the same operations to both sides of an inequality.
  • Reverse the inequality symbol when multiplying or dividing by a negative.
  • Solve and represent compound inequalities.
  • Split absolute value inequalities into two separate linear inequalities.

Example 1: Solving a Simple Inequality

Solve for xx:
2x−5<72x - 5 < 7
Step 1: Add 5 to both sides.
2x<122x < 12
Step 2: Divide by 2.
x<6x < 6
Final Answer: x<6\boxed{x < 6} Graph: Shade all numbers less than 6 on the number line (open circle at 6).

Example 2: Absolute Value Inequality

Solve for xx:
∣x−3∣≤5|x - 3| \le 5
Step 1: Write as a compound inequality.
−5≤x−3≤5-5 \le x - 3 \le 5
Step 2: Add 3 to all sides.
−2≤x≤8-2 \le x \le 8
Final Answer: −2≤x≤8\boxed{-2 \le x \le 8} Interpretation: All values of xx that are within 5 units of 3 satisfy the inequality.

Key Takeaways

  • Solving inequalities follows the same steps as solving equations, except when multiplying or dividing by a negative number - always flip the sign.
  • Compound inequalities combine solution sets with “and” or “or”.
  • Absolute value inequalities describe distances from a central point on the number line.