Algebra Chapter 5: Solving Inequalities
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Questions and Answers

What is the solution to the inequality $-2x \geq 4$?

  • $x \leq -2$ (correct)
  • $x \leq 2$
  • $x \geq 2$
  • $x \geq -2$

    • What is the largest prime number value of $x$ that satisfies the inequality $8 - x > 3$?

  • 5 (correct)
  • 2
  • 3
  • 4

    • What is the smallest rational value of $x$ that satisfies the inequality $3x \leq 8x - 5$?

  • 2
  • 0.5
  • 1.5 (correct)
  • 3.5

    • For the inequality $5(x + 3) \geq 9$, what is the solution for $x$?

    <p>$x \geq -3$</p> Signup and view all the answers

    What is the smallest possible value of $x$ that is a perfect square and satisfies $-4y - 5 \geq 11$?

    <p>4</p> Signup and view all the answers

    In the equation $2x + y = 2$, when $x = -4$, the value of $y$ is _____

    <p>-10</p> Signup and view all the answers

    In the equation $2x + y = 2$, when $x = 0$, the value of $y$ is _____

    <p>2</p> Signup and view all the answers

    In the equation $2x + y = 2$, when $x = 4$, the value of $y$ is _____

    <p>-6</p> Signup and view all the answers

    What is the value of p if the point (p, -2) lies on the graph of $2x + y = 2$?

    <p>2</p> Signup and view all the answers

    The coordinates of the point of intersection of the graphs of $2x + y = 2$ and $x = -0.5$ are _____

    <p>(-0.5, 3)</p> Signup and view all the answers

    Find the value of p when $5x - 3y = 2$ and $x = -5$.

    <p>7</p> Signup and view all the answers

    If $5x - 3y = 2$ and $x = -2$, what is the value of y?

    <p>-4</p> Signup and view all the answers

    Find the value of q when $5x - 3y = 2$ and $x = 7$.

    <p>11</p> Signup and view all the answers

    What are the simultaneous equations to solve when $5x - 3y = 2$ and $3x + 4y = 7$?

    <p>x = 2, y = 1</p> Signup and view all the answers

    What is the first number if 11 is added to it, resulting in a number twice the second number?

    <p>x + 11 = 2y</p> Signup and view all the answers

    What is the difference in price between 1 cup of ice-cream milk tea and 1 cup of bubble tea if 5 cups and 4 cups costs $26.80 and 7 cups and 6 cups costs $38.60?

    <p>$2.50</p> Signup and view all the answers

    Study Notes

    Solving Inequalities

    • Solve the inequality -2x ≥ 4

      • Divide both sides of the inequality by -2, remembering to flip the inequality sign
      • The solution is x ≤ -2.
      • Represent this solution on a number line with a closed circle at -2 and a line extending to the left.

    • Solve the inequality x + 4 < 3

      • Subtract 4 from both sides of the inequality.
      • The solution is x < -1.
      • Represent this solution on a number line with an open circle at -1 and a line extending to the left.

    • Solve the inequality -4y - 5 ≥ 11

      • Add 5 to both sides of the inequality.
      • Divide both sides of the inequality by -4, remembering to flip the inequality sign.
      • The solution is y ≤ -4.
      • Represent this solution on a number line with a closed circle at -4 and a line extending to the left.

    • Solve the inequality -2y + 4 > 3

      • Subtract 4 from both sides of the inequality.
      • Divide both sides of the inequality by -2, remembering to flip the inequality sign.
      • The solution is y < 1/2.
      • Represent this solution on a number line with an open circle at 1/2 and a line extending to the left.

    • Solve the inequality x - 3 ≥ 7

      • Add 3 to both sides of the inequality.
      • The solution is x ≥ 10.
      • Represent this solution on a number line with a closed circle at 10 and a line extending to the right.

    Solving Inequalities with Conditions

    • Solve the inequality 8 - x > 3.

      • Subtract 8 from both sides of the inequality.
      • Multiply both sides of the inequality by -1, remembering to flip the inequality sign.
      • The solution is x < 5.
      • Represent this solution on a number line with an open circle at 5 and a line extending to the left.

    • Find the largest possible value of x that satisfies the inequality 8 - x > 3 if x is a prime number

      • The largest prime number less than 5 is 3

    • Find the positive integer value of x that satisfies the inequality 8 - x > 3.

      • The positive integer that satisfies the inequality is 4.

    • Find the largest possible value of x that satisfies the inequality 8 - x > 3 if x is a perfect square.

      • The largest perfect square that satisfies the inequality is 4.

    Additional Inequality Problems

    • Solve the inequality 3x ≤ 8x - 5.

      • Subtract 3x from both sides of the inequality.
      • Divide both sides by 5.
      • The solution is x ≥ 1.
      • Represent this solution on a number line with a closed circle at 1 and a line extending to the right.

    • Find the smallest possible value of x that satisfies the inequality 3x ≤ 8x - 5 if x is a prime number.

      • The smallest prime number that satisfies the inequality is 2.

    • Find the smallest rational value of x that satisfies the inequality 3x ≤ 8x - 5.

      • The smallest rational value that satisfies the inequality is 1.

    • Find the smallest perfect cube that satisfies the inequality 3x ≤ 8x - 5.

      • The smallest perfect cube that satisfies the inequality is 1.

    Solving More Complex Inequalities

    • Solve the inequality 5(x + 3) ≥ 9.

      • Distribute the 5 on the left side of the inequality.
      • Subtract 15 from both sides.
      • Divide both sides by 5.
      • The solution is x ≥ -6/5.
      • Represent this solution on a number line with a closed circle at -6/5 and a line extending to the right.

    • Solve the inequality 𝑦+1/3 > 1/2.

      • Subtract 1/3 from both sides of the inequality.
      • The solution is y > 1/6.
      • Represent this solution on a number line with an open circle at 1/6 and a line extending to the right.

    Linear Equations

    • Solve for y in the equation 2x + y = 2 to create a table of values for x and y

    • Graphing Linear Equations

    • Draw the graph of 2x + y = 2 for -4 ≤ x ≤ 4 on a sheet of graph paper using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis

    • Identify the point (p, -2) on the graph and find the value of p

    • Draw the graph of x = -0.5 on the same axes

    • Find the coordinates of the point where the graphs of 2x + y = 2 and x = -0.5 intersect

    • Solving Systems of Equations

    • Find the values of p and q in the equation 5x - 3y = 2 using the table of values provided

    • Graph the equation 5x - 3y = 2 for -5 ≤ x ≤ 7 on a sheet of graph paper using a scale of 1 cm to represent 1 unit on both axes

    • Create a table of values for x and y in the equation 3x + 4y = 7

    • Graph the equation 3x + 4y = 7 on the same axes as the previous graph

    • Find the solution to the system of equations 5x - 3y = 2 and 3x + 4y = 7 by identifying the point of intersection on the graph

    • Simultaneous Equations

    • Solve the simultaneous equations 7x + 2y = 10 and 5x - 4y = 28

    • Solve the simultaneous equations 5x + 2y = 6 and 2x - y = 2.5

    • Word Problems

    • Let the first number be x and the second number be y

    • Formulate two equations based on the given information: - x + 11 = 2y - y + 20 = 2x

    • Solve the simultaneous equations to find the two numbers

    • Real-World Problems

    • Let the cost of 1 cup of ice-cream milk tea be x and the cost of 1 cup of bubble tea be y

    • Formulate two equations based on the given information: - 5x + 4y = 26.80 - 7x + 6y = 38.60

    • Solve the simultaneous equations to find the difference between x and y

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    Description

    This quiz focuses on solving inequalities and representing solutions on number lines. It includes various types of inequalities, demonstrating the process of isolating the variable and graphing the results. Test your knowledge and skills in algebraic problem-solving.

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