Lesson 4: Linear Equations and Inequalities

New Concept

This lesson expands your equation-solving skills to multi-step problems. You'll learn to isolate variables by reversing operations, a key strategy for solving both linear equations and inequalities, and apply these methods to real-world scenarios.

What’s next

Now, let's master this by tackling guided examples. You'll then apply these steps in practice problems and solve inequalities with interactive exercises.

Property

We solve an equation by isolating the variable on one side of the equation. If an equation involves two or more operations, we must undo those operations in reverse order.

Examples

• To solve $4x + 7 = 23$, first subtract 7 from both sides to get $4x = 16$. Then, divide both sides by 4 to find $x = 4$.
• To solve $15 - 2y = 5$, first subtract 15 from both sides to get $-2y = -10$. Then, divide both sides by $-2$ to find $y = 5$.
• To solve $\frac{z}{5} - 3 = 2$, first add 3 to both sides to get $\frac{z}{5} = 5$. Then, multiply both sides by 5 to find $z = 25$.

Explanation

Think of this as reversing the order of operations. To isolate the variable, you first undo any addition or subtraction, then you undo any multiplication or division. It is like taking off your shoes before your socks to get to your feet.

Property

A statement that uses one of the symbols $>$ or $<$ is called an inequality. An inequality that uses the symbol for less than, $<$, or greater than, $>$, is called a strict inequality. A nonstrict inequality uses one of the following symbols: $\geq$ means "greater than or equal to"; $\leq$ means "less than or equal to".

Examples

• The inequality $x > 5$ represents all numbers strictly greater than 5. On a number line, this is shown with an open circle at 5 and an arrow pointing to the right.
• The inequality $y \leq -2$ represents -2 and all numbers less than it. On a number line, this is shown with a solid dot at -2 and an arrow pointing to the left.
• The values 8, 9.5, and 200 all satisfy the inequality $x \geq 8$, but 7.9 does not.

Explanation

Inequalities describe a range of possible values, not just a single answer. A strict inequality ($<$ or $>$) uses an open circle on a number line, while a non-strict one ($\leq$ or $\geq$) uses a solid dot to show the endpoint is included.

Property

To solve an inequality:

1. We can add or subtract the same quantity on both sides.
2. We can multiply or divide both sides by the same positive number.
3. If we multiply or divide both sides by a negative number, we must reverse the direction of the inequality.

Examples

• To solve $2x - 5 < 9$, add 5 to both sides to get $2x < 14$. Then, divide by 2 to get $x < 7$. The inequality sign does not change.
• To solve $14 - 4x \geq 6$, subtract 14 to get $-4x \geq -8$. Then, divide by $-4$ and reverse the inequality sign to get $x \leq 2$.
• To solve $2 - \frac{x}{3} > 4$, subtract 2 to get $-\frac{x}{3} > 2$. Then, multiply by $-3$ and reverse the inequality sign to get $x < -6$.

Explanation

Solving an inequality is just like solving an equation, with one crucial exception. Remember this golden rule: if you multiply or divide both sides by a negative number, you MUST flip the direction of the inequality sign ($<$ becomes $>$, and vice versa).

Property

An inequality in which the variable expression is bounded from above and from below is called a compound inequality. To solve a compound inequality, we must perform the steps needed to isolate x on all three sides of the inequality.

Examples

• To solve $-2 < 3x + 4 \leq 16$, first subtract 4 from all three parts: $-6 < 3x \leq 12$. Then, divide all parts by 3: $-2 < x \leq 4$.
• To solve $1 \leq 7 - 2x < 9$, first subtract 7 from all three parts: $-6 \leq -2x < 2$. Then, divide by $-2$ and reverse both inequality signs: $3 \geq x > -1$.
• To solve $0 < \frac{x+2}{5} < 3$, first multiply all three parts by 5: $0 < x+2 < 15$. Then, subtract 2 from all parts: $-2 < x < 13$.

Explanation

A compound inequality is like a sandwich where the variable is trapped between two values. To solve it, whatever operation you do to the middle part to isolate the variable, you must also do to both of the outside numbers.