Graph each inequality. 4y - 3x ≤ 5

Verified step by step guidance

1. Rewrite the inequality in slope-intercept form (y = mx + b) by solving for y. Start with the inequality: 4y - 3x \leq 5.
2. Add 3x to both sides to isolate the term with y: 4y \leq 3x + 5.
3. Divide every term by 4 to solve for y: y \leq \frac{3}{4}x + \frac{5}{4}.
4. Graph the line y = \frac{3}{4}x + \frac{5}{4} as a solid line because the inequality is \leq (less than or equal to).
5. Shade the region below the line to represent the solution to the inequality y \leq \frac{3}{4}x + \frac{5}{4}.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Understanding how to interpret and manipulate inequalities is essential for graphing them accurately.

Graphing Linear Inequalities

Graphing linear inequalities involves representing the solutions of the inequality on a coordinate plane. The boundary line is drawn based on the corresponding equation, and it is solid if the inequality includes equality (≤ or ≥) and dashed if it does not (< or >). The region that satisfies the inequality is then shaded to indicate all possible solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b represents the y-intercept. This form is useful for graphing because it allows you to easily identify the starting point on the y-axis and the angle of the line. Converting the given inequality into this form can simplify the graphing process.