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Mom's Basement University

Simple Inequalities

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what inequalities are, how to solve them using inverse operations, and when to flip the inequality sign.

The Concept

Section titled “The Concept”

An inequality compares two values using < (less than), > (greater than), ≤ (less than or equal), or ≥ (greater than or equal).

Solving inequalities works like equations (use inverse operations to isolate the variable) but with one key rule:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Examples:

x+5>12x>7(no flip)\begin{aligned} x + 5 &> 12 \\ x &> 7 \quad \text{(no flip)} \end{aligned} 3x≤18x≤6(no flip)\begin{aligned} 3x &\leq 18 \\ x &\leq 6 \quad \text{(no flip)} \end{aligned} −2x<10x>−5(flip: divided by −2)\begin{aligned} {-}2x &< 10 \\ x &> {-}5 \quad \text{(flip: divided by }{-}2\text{)} \end{aligned} 4−y≥9−y≥5y≤−5(flip: multiplied by −1)\begin{aligned} 4 - y &\geq 9 \\ {-}y &\geq 5 \\ y &\leq {-}5 \quad \text{(flip: multiplied by }{-}1\text{)} \end{aligned}

Graphing: Use an open circle (○) for < or >, closed circle (●) for ≤ or ≥, and shade the direction of the solution.

Worked Examples

Section titled “Worked Examples”

Solve and graph each:

1. 2x−7<52x - 7 < 5

2x−7<52x<12x<6\begin{aligned} 2x - 7 &< 5 \\ 2x &< 12 \\ x &< 6 \end{aligned}

Graph: open circle at 6, shade left (all numbers less than 6).

2. −3y+4≥13-3y + 4 \geq 13

−3y+4≥13−3y≥9y≤−3(flip: divided by −3)\begin{aligned} {-}3y + 4 &\geq 13 \\ {-}3y &\geq 9 \\ y &\leq {-}3 \quad \text{(flip: divided by }{-}3\text{)} \end{aligned}

Graph: closed circle at −3-3, shade left (all numbers less than or equal to −3-3).

Check: Pick a test point in the shaded area and verify it satisfies the original inequality.

Real-World Applications

Section titled “Real-World Applications”

Inequalities describe ranges and limits in everyday life:

  • Budget: You can spend less than 200 dollars on groceries → g<200g < 200
  • Work: Need at least 40 hours for full benefits → h≥40h \geq 40
  • Speed limit: Drive no faster than 65 mph → s≤65s \leq 65
  • Savings goal: Need more than 500 dollars for a trip → s>500s > 500

These help set boundaries, compare options, and make decisions about money, time, or resources.

Quiz

Section titled “Quiz”
Solve $x + 8 > 15$.
Solve $-4y \leq 12$.
You need more than 50 dollars for a gift. Write the inequality.
Solve $6 - 2z < 4$.
Solve $3x + 7 \leq 22$.