Linear Equations And Inequalities

What are linear equations?

• A linear equation can be written in the form $ax + b = c$, where a, b, c are numbers and x is the variable.
• The highest power of x is $1 (no x^{2} terms)$.
• Examples: $2x + 3 = 5, 3x + 4 = 1, x - 5 = -3$.

Which of the following is a linear equation?

• $x^{2} + 2 = 5$
• $3x - 1 = 7$
• $x + \frac{1}{x} = 2$
• $\sqrt{x} = 4$

Solving linear equations: inverse operations

• To solve, use inverse operations to isolate x.
• Addition and subtraction are inverse operations; multiplication and division are inverse operations.
• Whatever you do to one side, do to the other.
• Example: $2x + 1 = 9$ → subtract 1 → $2x = 8$ → divide by 2 → $x = 4$.

To solve $3x - 5 = 10$, first add ____ to both sides, then divide by ____.

Solving equations with brackets

• If an equation has brackets, expand them first.
• Example: $2(x - 3) = 10$ → expand: $2x - 6 = 10$ → add $6: 2x = 16$ → divide by $2: x = 8$.
• Alternatively, you can divide both sides by the number outside the bracket first, but be careful with fractions.

Solve $3(x + 2) = 15$.

• $x = 3$
• $x = 5$
• $x = 7$
• $x = 13$

Solving equations with x on both sides

• Collect x terms on one side by adding or subtracting the smaller x term.
• Example: $4x - 7 = 11 + x$ → subtract $x: 3x - 7 = 11$ → add $7: 3x = 18$ → divide by $3: x = 6$.
• This avoids negative coefficients.

Solve $5x + 3 = 2x + 12$: subtract ____ from both sides, then subtract 3, then divide by ____.

Solving equations with fractions

• Multiply both sides by the lowest common denominator to eliminate fractions.
• Example: $\frac{x}{5} + 4 = \frac{9}{2}$ → multiply by $10: 2x + 40 = 45$ → subtract $40: 2x = 5$ → $x = \frac{5}{2}$.
• If the unknown is in the denominator, multiply both sides by that denominator.

Solve $\frac{x}{3} + 1 = 5$.

• $x = 12$
• $x = 18$
• $x = 2$
• $x = 4$

What are inequalities?

• Inequalities show that something is greater (>) or less (<) than something else.
• ≥ means greater than or equal to; ≤ means less than or equal to.
• Strict inequalities: $> and < do$ not include the endpoint.
• Example: $x > 5$ means x can be 6, 7, ... but not 5.

Which of these are true?

• $x \ge 3$ includes $x = 3$.
• $x < 2$ includes $x = 2$.
• $x > 0$ includes $x = 0$.
• $x \le -1$ includes $x = -1$.

Finding integer values from inequalities

• List all integers between the endpoints, checking if endpoints are included.
• Example: $3 \le x < 6$ gives $x = 3, 4, 5 (3$ included, 6 not).
• If only one endpoint, there are infinitely many integers $(e.g., x > 2$ gives 3, 4, 5, ...).
• Remember zero and negative integers are included unless stated otherwise.

Representing inequalities on a number line

• Use an open circle $for < or > (endpoint$ not included).
• Use a closed (filled) circle $for \le or \ge (endpoint included)$.
• Draw a line connecting the circles, or an arrow for one-sided inequalities.
• Example: $-3 < x \le 4$: open circle at -3, closed at 4, line between.

Which number line represents $x \ge 2$?

• Closed circle at 2, arrow right
• Open circle at 2, arrow right
• Closed circle at 2, arrow left
• Open circle at 2, arrow left

Solving linear inequalities

• Solve inequalities the same way as equations, keeping the inequality sign.
• When multiplying or dividing by a negative number, flip the inequality sign.
• Example: $-2x < 6$ → divide $by -2: x > -3 (sign flips)$.
• Never multiply or divide by a variable (x) as its sign may be unknown.

Solve $3x - 7 \le 8$: add ____ to both sides, then divide by ____, giving x \le ____.

Solving double inequalities

• Apply the same operation to all three parts.
• Example: $a < 2x < b$ → divide by $2: \frac{a}{2} < x < \frac{b}{2}$.
• Alternatively, split into two separate inequalities and find the overlap.

Solve $-3 < 2x \le 8$.

• $-1.5 < x \le 4$
• $-3 < x \le 4$
• $-1.5 < x \le 8$
• $-3 < x \le 8$