Mathematics 8 - Linear Equations and Inequalities

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Questions and Answers

What is the standard form of a linear equation in two variables?

  • Ax + By < C
  • Ax + By = C (correct)
  • Ax + By > C
  • Ax + By ≥ C

Which symbol represents a linear inequality that is less than or equal to?

  • <
  • ≤ (correct)
  • ≥
  • >

What is the result of shading the region above the boundary line when graphing an inequality like y ≥ mx + b?

  • It indicates values are greater than the boundary line. (correct)
  • It indicates the boundary line is irrelevant.
  • It indicates values are less than the boundary line.
  • It indicates both equal and unequal values.

In the context of graphing linear inequalities, what type of line is drawn for inequalities using the less than (<) or greater than (>) symbols?

<p>Dashed line (B)</p> Signup and view all the answers

When rewriting the inequality 3x + 2y < 6, what should be solved for to graph it?

<p>y (B)</p> Signup and view all the answers

What is the primary difference between linear equations and linear inequalities?

<p>Linear equations use an equal sign; linear inequalities use inequality symbols. (B)</p> Signup and view all the answers

Which of the following statements about the graph of linear inequalities is true?

<p>It shows the complete solution set within half-planes. (D)</p> Signup and view all the answers

Why are A and B in the forms of linear equations and inequalities not equal to zero?

<p>To avoid undefined slopes. (B)</p> Signup and view all the answers

What type of inequality can be expressed in the form $Ax + By < C$, $Ax + By > C$, $Ax + By ≤ C$, or $Ax + By ≥ C$?

<p>Linear Inequality (C)</p> Signup and view all the answers

Which of the following represents a Linear Inequality in Two Variables?

<p>$-5y ≥ 3$ (A), $2y ≤ 6$ (B), $7x - 4y &gt; 3$ (D)</p> Signup and view all the answers

Which of the following best describes the graph of Linear Inequalities in Two Variables?

<p>Half Plane (C)</p> Signup and view all the answers

A Linear Inequality in Two Variables can have how many solutions?

<p>Infinite (D)</p> Signup and view all the answers

What is the name of the line that separates the plane into two regions in Linear Inequalities?

<p>Plane Divider (C)</p> Signup and view all the answers

Which of the following is NOT a Linear Inequality in Two Variables?

<p>$4 + 8x &lt; 14y$ (A)</p> Signup and view all the answers

When will an inequality have a solid line as a plane divider?

<p>When the symbol includes 'or equal to' (C)</p> Signup and view all the answers

What type of line should be drawn for the inequality y ≤ 2x – 1?

<p>Solid line (C)</p> Signup and view all the answers

Which points are considered solutions for the inequality y ≤ 2x – 1?

<p>(2, 3) (A), (4, 1) (C)</p> Signup and view all the answers

When rewriting the inequality 2y – x ≤ 6, what is the first step?

<p>Add x to both sides (B)</p> Signup and view all the answers

What does the inequality symbol '≤' indicate regarding the solutions of the boundary line?

<p>Points on and below the line are solutions (B)</p> Signup and view all the answers

How do you determine the shaded region for an inequality graph?

<p>By shading below the boundary line for '≤' (C)</p> Signup and view all the answers

What is the solution set of the inequality 2y – x ≤ 6?

<p>y ≤ (1/2)x + 3 (D)</p> Signup and view all the answers

Given the points plotted (-2, 2), (0, 3), and (2, 4), which point specifically does NOT satisfy y ≤ 2x – 1?

<p>(2, 4) (B)</p> Signup and view all the answers

What is achieved by dividing both sides of the inequality 2y ≤ x + 6 by 2?

<p>It simplifies to y ≤ (1/2)x + 3 (B)</p> Signup and view all the answers

Which of the following equations is categorized as a Linear Inequality in Two Variables?

<p>2𝑥 − 3𝑦 ≥ 9 (B)</p> Signup and view all the answers

What is the first step to graph the inequality 𝑦 ≤ 2𝑥 - 1?

<p>Rewrite the inequality to isolate ‘y’ (B)</p> Signup and view all the answers

Which of the following points is included in the solution set of the inequality 𝑦 ≤ 2𝑥 - 1?

<p>(0, -1) (A)</p> Signup and view all the answers

What is the correct value of ‘y’ when x = 1 in the equation y = 2𝑥 - 1?

<p>1 (B)</p> Signup and view all the answers

When graphing the inequality 𝑦 > -𝑥 - 5, the boundary line is plotted as which type?

<p>Dashed line (A)</p> Signup and view all the answers

In the inequality 3𝑦 + 6𝑥 < 0, what can be said about the corresponding boundary line?

<p>It is dashed. (A)</p> Signup and view all the answers

Which of the following must be true for a line representing the inequality 2𝑥 - 3𝑦 ≥ 9?

<p>The line is solid. (D)</p> Signup and view all the answers

What is the significance of shading below the plane divider in the inequality represented by 'y ≤ 2 + 3'?

<p>It shows the solutions where the inequality is satisfied. (B)</p> Signup and view all the answers

Which of the following points is a solution to the inequality '2y - x ≤ 6'?

<p>(4, 2) (A)</p> Signup and view all the answers

What does the inequality 'y + 2 > x' imply about the relationship between y and x?

<p>y can exceed x by any amount greater than 2. (D)</p> Signup and view all the answers

What is the first step in rewriting the inequality 'y + 2 > x'?

<p>Subtract 2 from both sides. (A)</p> Signup and view all the answers

If the boundary line of an inequality is solid, what does it indicate about the solutions?

<p>Solutions include points on the line. (B)</p> Signup and view all the answers

What does the statement 'y > 2x - 4' indicate about the solutions?

<p>The solutions are above the line. (B)</p> Signup and view all the answers

Which of the following is a step to graph the inequality 'y > 2x - 4'?

<p>All of the above. (D)</p> Signup and view all the answers

Which of the following points is a solution to the inequality y > 2x - 4?

<p>(2, 3) (B)</p> Signup and view all the answers

Why are the points (1,1), (2,3), and (1,2) not solutions to the given inequality?

<p>They are located on the boundary line. (D)</p> Signup and view all the answers

Which point does not satisfy the inequality '2y - x ≤ 6'?

<p>(5, 8) (D)</p> Signup and view all the answers

When performing operations on inequalities, what must you remember?

<p>You must reverse the inequality sign when multiplying or dividing by a negative number. (C)</p> Signup and view all the answers

What is the correct representation of the inequality when the values are shaded?

<p>y &gt; 2x - 4 (A)</p> Signup and view all the answers

Which of the following is NOT a characteristic of the inequality y > 2x - 4?

<p>The boundary points are included. (D)</p> Signup and view all the answers

Given the inequality 5x - y < 2, what is the nature of the graph?

<p>It is a diagonal line with dashed boundary. (A)</p> Signup and view all the answers

For which inequality does the solution include the boundary line?

<p>y ≥ 9x + 3 (A)</p> Signup and view all the answers

What is the graphical representation of 'x ≤ 3y + 5'?

<p>Shaded region below the line. (D)</p> Signup and view all the answers

Flashcards

Linear Equation in Two Variables

An equation of the form Ax + By = C, where A, B, and C are numbers and A and B are not both zero.

Linear Inequality in Two Variables

An inequality of the form Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C, where A, B, and C are numbers and A and B are not both zero.

Graphing Linear Inequalities

Representing linear inequalities in a coordinate plane by shading a region that includes all points satisfying the inequality.

Boundary Line

The line representing the equality part of the inequality.

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Solid Line

Used for inequalities with "≤" or "≥" (equal to or less than/greater than).

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Dashed Line

Used when an inequality has "<", or ">".

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Half-Plane

One of the two regions created by graphing a linear equation in a coordinate plane.

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Graphing Inequality Steps

Steps to graph a linear inequality in two variables - rewrite to isolate y, graph boundary, and shade the correct half-plane.

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Solution Set (Linear Inequality)

The region that satisfies the linear inequality.

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Solving a linear inequality

Finding all the solutions to an inequality involving two variables—essentially, determining the region on a coordinate plane that satisfies the inequality.

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Inequality Symbol

A symbol used to show the relationship between values: ≠, <, >, ≤, ≥.

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Graphing Steps (Linear Inequality)

A step-by-step process to graph a linear inequality involving two variables.

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Shading for 'y ≤'

When graphing an inequality with 'y ≤', shade the area below the boundary line.

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Shading for 'y ≥'

When graphing an inequality with 'y ≥', shade the area above the boundary line.

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Solid Line vs. Dashed Line

A solid line is used when the inequality includes '≤' or '≥'. A dashed line is used for '<' or '>' to indicate the boundary isn't included in the solution.

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Finding Solutions on a Graph

Any point within the shaded region on the graph of a linear inequality is a solution. Points on the solid boundary are also solutions. Points outside the shaded region or on a dashed boundary are not solutions.

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Isolating 'y' in an Inequality

Before graphing a linear inequality, rewrite it so that 'y' is alone on the left side of the inequality sign. This helps find points for the boundary line.

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Boundary Line Equation

The equation of the line that represents the equality part of an inequality is called the boundary line equation.

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What does a solid line mean in a linear inequality graph?

It indicates that the points on the line are included in the solution set of the inequality. This occurs when the inequality involves '≤' or '≥'.

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How do you determine the shaded region in a linear inequality graph?

After graphing the boundary line, shade the area above or below the line based on the inequality symbol. If the inequality is 'y ≥ ...', shade above the line. If the inequality is 'y ≤ ...', shade below the line.

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Why is it important to rewrite the inequality in terms of 'y'?

Rewriting the inequality in 'y = ...' form makes it easier to graph the boundary line. By isolating 'y', you can easily determine the slope and y-intercept of the line,

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How to graph a linear inequality

  1. Rewrite the inequality in 'y = ...' form.
  2. Graph the boundary line (solid for '≤' or '≥', dashed for '<' or '>')
  3. Shade the area above the line for 'y ≥ ...', or below the line for 'y ≤ ...'.
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Why are some points solutions and others are not?

Points that lie within the shaded region of the graph represent solutions to the inequality because they satisfy the inequality. Points outside the shaded region do not satisfy the inequality and therefore are not solutions.

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Example of a linear inequality

2y - x ≤ 6

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Solution points

Points that satisfy the inequality and are located within the shaded region. For example: (2, 4), (0, 3), (-2, 2)

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Shading for 'y >'

When graphing an inequality with 'y >', shade the area above the boundary line.

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Shading for 'y <'

When graphing an inequality with 'y <', shade the area below the boundary line.

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Solutions of a Linear Inequality

The set of all points that satisfy a linear inequality. They are represented by the shaded region and, if applicable, the solid boundary line.

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Linear Inequality

An inequality that can be written in the form Ax + By < C , Ax + By > C , Ax + By ≤ C or Ax + By ≥ C where A, B, and C are real numbers, and A and B are not both zero.

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Shading in Linear Inequality Graphs

The process of marking a region on a graph to indicate the set of solutions for a linear inequality. It depends on the direction of the inequality symbol.

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Solution Set

The set of all points that satisfy a given linear inequality, represented by the shaded region on a coordinate plane.

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Study Notes

Mathematics 8 - Learning Activity Sheet #1 (2nd Quarter - Week 1)

  • This document outlines concepts related to linear equations and inequalities in two variables.
  • Learning Competency: Differentiates linear inequalities in two variables from linear equations in two variables, and illustrates/graphs linear inequalities in two variables.
  • Background:
    • A linear equation in two variables is of degree one and written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero.
    • A linear inequality in two variables can be written as Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C, where A, B, and C are real numbers, and A and B are not both zero. Linear inequalities use < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Inequality statements differ from equations by not using an equal sign.
  • Graphing Linear Inequalities in Two Variables:
    • Step 1: Rewrite the inequality so 'y' is isolated on one side.
    • Step 2: Graph the boundary line. A solid line is used for '≤' or '≥'; a dashed line is used for '<' or '>'.
    • Step 3: Shade the appropriate region. Shade above the line for '>' or '≥'; shade below the line for '<' or '≤'. The shaded area represents the solution set in the given inequality.
  • Activity Examples:
    • Examples show how to determine whether an equation is a linear equation or linear inequality.
    • Detailed steps on how to graph a linear inequality.
  • References and Resources: External links to helpful online resources concerning linear inequalities in two variables are provided.
  • Additional Activity (for study purposes): Multiple-choice questions on linear equations and inequalities in two variables. Covers topics such as determining whether something is linear equation or inequality, determining how to graph lines and inequalities, identifying solutions to inequalities, and the appropriate symbols used in inequalities.

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