Lesson 8-4: Describe Solutions to Linear Equations

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

An equation has infinitely many solutions when algebraic manipulation results in a true statement where both sides are identical, such as $a = a$ where $a$ is any real number.

Examples

Property

If no value of the variable makes an equation true, then the equation has no solution.

Examples

$7x - 2 = 7x - 5$ simplifies to $-2 = -5$. This is never true, so there is no solution.
$3(x + 4) = 3x + 10$ simplifies to $3x + 12 = 3x + 10$, which becomes $12 = 10$. Impossible! No solution.

Explanation

This is an equation that’s impossible to solve! No matter what number you try, it won't work. After you simplify, the variables disappear, leaving a false statement like $-2 = -5$. The equation is telling you, 'Nope, not happening!' There's just no answer that can make the equation true.