Eighth Grade Mathematics: Solving and Graphing Multi-Step Inequalities
While an equation states that two expressions are equal, an inequality describes a condition where one expression is greater than, less than, or bounded by another. In the physical world, speed limits, minimum wage requirements, load capacities of bridges, and budget constraints are all defined by inequalities. Solving a linear inequality follows almost the same balancing rules as solving an equation, with one profound exception: whenever you multiply or divide both sides by a negative number, the direction of the inequality symbol must reverse. In this chapter, you will master solving multi-step inequalities and graphing their solution sets on the number line.
Inequality Symbols and Number Line Conventions
Inequalities express relationships of order using four standardized relational symbols:
Symbol Meaning Number Line Circle Type
-----------------------------------------------------------------
< Strictly Less Than Open Circle (Value not included)
> Strictly Greater Than Open Circle (Value not included)
<= Less Than or Equal To Closed Circle (Value is included)
>= Greater Than or Equal To Closed Circle (Value is included)
Graphing x > 3 (Open circle at 3, shaded to the right):
<----+----+----+----+----+----o====*====*====*====>
-1 0 1 2 3 4 5 6 7
Graphing x <= -2 (Closed circle at -2, shaded to the left):
<====*====*====*====*----*----+----+----+----+---->
-5 -4 -3 -2 -1 0 1 2 3
An open circle indicates a strict boundary: the number itself does not satisfy the inequality, but numbers infinitely close to it do. A closed (solid) circle indicates that the boundary value is an acceptable solution.
The Negative Multiplication and Division Rule
The defining operational rule of inequalities states: when multiplying or dividing both sides of an inequality by a negative number, you MUST reverse the direction of the inequality symbol.
Why Does the Symbol Flip?
Start with a known true statement: 2 < 5
Multiply both sides by -1: -1 * (2) ? -1 * (5)
-2 ? -5
On the number line, -2 is to the RIGHT of -5:
-2 > -5 (The symbol reversed!)
Multiplying by a negative reflects values across the origin zero to the opposite side of the number line, reversing their relative order. If you forget to reverse the symbol, your entire solution set will point in the wrong direction.
Solving Multi-Step Inequalities
Multi-step inequalities are solved using the same simplification sequence as multi-step equations: distribute to remove parentheses, combine like terms on each side, isolate the variable terms, and isolate the variable.
Solve: -3*(2x - 4) + 5 >= 29
Step 1: Distribute -3:
-6x + 12 + 5 >= 29
Step 2: Combine constants on the left:
-6x + 17 >= 29
Step 3: Subtract 17 from both sides:
-6x >= 12
Step 4: Divide both sides by -6 (REVERSE the inequality symbol!):
x <= 12 / (-6)
x <= -2
The symbol flipped from >= to <= because we divided by the negative coefficient -6.
Chapter Exercises
Exercise 1: Solve the multi-step inequality 4*(x - 2) - 7x < 13. State the solution inequality, explain how the negative sign rule applies during the final division step, and describe how to graph the solution on a number line.
Exercise 2: Solve the inequality 2*(5x + 3) >= 7x - 12. Show all algebraic steps and check whether x = -6 is a solution.
Exercise 3: A freight elevator has a maximum safe carrying capacity of 2400 pounds. An elevator operator weighing 180 pounds needs to transport identical cargo crates weighing 75 pounds each. Write an inequality representing this scenario, solve for the maximum number of crates that can be safely transported in one trip, and explain your rounding decision.
Solutions and Explanations
Solution 1: To solve 4*(x - 2) - 7x < 13, begin by distributing 4 into (x - 2): 4x - 8 - 7x < 13. Next, combine like terms on the left side: (4x - 7x) - 8 = -3x - 8 < 13. Isolate the variable term by adding 8 to both sides: -3x < 21. Finally, divide both sides by -3. Because we are dividing by a negative number, the inequality symbol must reverse from less than (<) to greater than (>): x > 21 / (-3), which yields x > -7. To graph this solution on a number line, place an open circle at -7 to indicate that -7 is not included in the solution set, and draw a solid ray extending to the right with an arrow, covering all numbers greater than -7.
Solution 2: To solve 2(5x + 3) >= 7x - 12, distribute the 2 on the left side: 10x + 6 >= 7x - 12. Next, eliminate 7x from the right side by subtracting 7x from both sides: 3x + 6 >= -12. Subtract 6 from both sides to isolate the variable term: 3x >= -18. Divide both sides by positive 3. Because 3 is positive, the inequality symbol does NOT flip: x >= -18 / 3, which yields x >= -6. To check whether x = -6 is a solution, substitute -6 into the original inequality: 2(5(-6) + 3) >= 7(-6) - 12. Left side: 2(-30 + 3) = 2(-27) = -54. Right side: -42 - 12 = -54. Since -54 >= -54 is true (because of the equal-to condition), x = -6 is indeed a valid solution.
Solution 3: Let c represent the number of cargo crates loaded onto the elevator. The total weight consists of the operator's weight (180 pounds) plus the combined weight of the crates (75c pounds). The total weight cannot exceed 2400 pounds, giving the inequality 75c + 180 <= 2400. To solve for c, subtract 180 from both sides: 75c <= 2220. Divide both sides by 75: c <= 29.6. In this physical context, you cannot load a fraction of a crate, and loading 30 crates would exceed the safe weight capacity (75*30 + 180 = 2430 pounds). Therefore, you must round down to the nearest whole integer regardless of the decimal value. The maximum number of crates that can be safely transported in a single trip is 29 crates.