5-2 Solving Multi-Step Inequalities

Property

Before you can begin moving terms across the inequality symbol, you must simplify each side independently. This is the "cleanup" phase:

• Step 1: Use the Distributive Property to remove any parentheses.
• Step 2: After distributing, combine any like terms on the same side of the inequality to finish simplifying the expression.

Examples

• Example 1 (Distributing): Simplify the inequality $8 - 2(x + 3) \leq 14$.

First, distribute the -2 to get $8 - 2x - 6 \leq 14$.
Then, combine the constant like terms (8 and -6) to get $-2x + 2 \leq 14$. Now it is ready to solve!

• Example 2 (Multiple Distributions): Simplify $4(x - 8) - (x + 3) > 10$.

Distribute the 4 and the negative sign: $4x - 32 - x - 3 > 10$.
Combine like terms ($4x$ with $-x$, and -32 with -3) to get $3x - 35 > 10$.

• Example 3: Simplify $4(3n + 7) + 5(n - 2) < 50$.

Distribute to get $12n + 28 + 5n - 10 < 50$.
Combine like terms to get $17n + 18 < 50$.

Property

To solve a multi-step linear inequality, follow a systematic flow:

1. Simplify each side completely (distribute and combine like terms).
2. Use the Addition or Subtraction Properties of Inequality to collect all variable terms on one side and all constant terms on the other side.
3. Use the Multiplication or Division Properties of Inequality to isolate the variable. (Remember to reverse the inequality sign if you multiply or divide by a negative number!)

Examples

• Example 1: Solve $3x + 5 > 20$.

Subtract 5 from both sides to get $3x > 15$.
Divide by 3 to get $x > 5$.

• Example 2 (Variables on both sides): Solve $7p - 2 \leq 3p + 10$.

Subtract $3p$ from both sides to gather variables on the left: $4p - 2 \leq 10$.
Add 2 to both sides to gather constants on the right: $4p \leq 12$.
Divide by 4 to get $p \leq 3$.

• Example 3 (Negative division): Solve $5(k - 2) > -20$.

Distribute to get $5k - 10 > -20$.
Add 10 to both sides: $5k > -10$.
Divide by 5 to get $k > -2$. (The sign stays the same because we divided by a positive 5).

Explanation

Solving a multi-step inequality uses the exact same strategy as solving a multi-step equation: clean up both sides, move the letters to one team and the numbers to the other, and then isolate the variable. The only difference is the golden rule of inequalities—you must stay highly alert during the very last step. If you divide or multiply by a negative number to get the variable by itself, you must flip the inequality symbol.

Property

When solving a multi-step inequality, sometimes the variable terms completely cancel each other out. When this happens, look at the remaining numerical statement:

• Contradiction (False Statement): If the statement is mathematically false (e.g., $5 < 2$), there is No Solution ($\emptyset$).
• Identity (True Statement): If the statement is mathematically true (e.g., $4 \geq 4$ or $-1 < 5$), the solution is All Real Numbers $(-\infty, \infty)$.

Examples

• Example 1 (No Solution): Solve $2(x + 3) \geq 2x + 9$.

Distribute: $2x + 6 \geq 2x + 9$.
Subtract $2x$ from both sides: $6 \geq 9$.
Since 6 is not greater than or equal to 9, this is a false statement. The solution set is $\emptyset$ (No Solution).

• Example 2 (All Real Numbers): Solve $3(x - 1) < 3x + 2$.

Distribute: $3x - 3 < 3x + 2$.
Subtract $3x$ from both sides: $-3 < 2$.
Since -3 is always less than 2, this is a true statement. The solution is All Real Numbers.

Explanation

Don't panic if your variable completely disappears while solving! When this happens, the math is telling you that the specific value of the variable doesn't actually matter. Your only job is to judge what is left. If the leftover numbers make a true statement, then literally any real number will work. If they make a false statement, then no number in the world can fix it.