text: 6_5 Graphing Linear Inequalities
  
  ```6.5 Slope intercept form
  for Inequalities:
  Linear Inequality: is a linear equation with
  an inequality sign (< , ≤, >, ≥)
  Solution of an Inequality: is an ordered
  pair (x, y) that makes the inequality true.
  GOAL:
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  Slope-Intercept Form: The linear equation of a
  nonvertical line with an inequality sign:
  y (<, ≤, >, ≥) m x + b
  Slope =
  −
  −
  =
  ⍙
  =
  ⍙
  y-intercept
  y crossing
  http://mathgraph.idwvogt.com/examples.html
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y<mx+b
  dash line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y>mx+b
  dash line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y≤mx+b
  Solid line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y≥mx+b
  Solid line
  EX:
  Provide the equation of the inequality.
  Solution: Since line is dashed and shaded at
  the bottom we use <. Also, the inequality
  must be in
  slope-intercept form:
  Y < mx + b
  1. Find the y-intercept
  In this graph b = +1.
  2. Find another point
  to get the slope.
  A(0,5)
  A(0,1)
  B(3,-2)
  Use the equation of slope to find the slope:
  −
  −
  =
  −−
  −
  =
  −
  
  = -1
  The slope-intercept
  form inequality is:
  A(0,1)
  B(3,-2)
  y < -1x + 1
  Remember:
  This means that if you start a 1 and move down
  one and over to the right one, and continue this
  pattern. We shade the bottom since it is <.
  When work does not need to be shown:
  (EOC Test)
  look at the triangle
  points.
  Count the number
  of square going up
  or down and to the
  right.
  In this case 1 down
  and 1 right. Thus slope is -1/1 = -1
  YOU TRY IT
  Provide the equation of the
  inequality.
  YOU TRY IT: (Solution)
  The inequality is solid and shaded below:
  Y ≤ mx + b
  1. Find the y-intercept
  In this graph b = + 4.
  2. Find another point
  to get the slope.
  A(0,4) B(1,0)
  A(0,4)
  B(1,0)
  Use the equation of slope to find the slope:
  −
  −
  =
  −
  −
  =
  −
  
  =-4
  The slope-intercept
  form equation is:
  A(0,4)
  B(1,0)
  y ≤ -4x + 4
  Remember:
  This means that if you start a 4 and move down
  four and one over to the right. Solid line and
  shaded down means we must use ≤.
  When no work is required, you can use the
  rise/run of a right triangle between the two
  points:
  Look at the triangle,
  down 4 (-4) over to the
  right 1 (+1)
  slope = -4/+1 = -4
  A(0,4)
  B(1,0)
  Remember:
  You MUST KNOW BOTH procedures, the
  slope formula and the triangle.
  Given Two Points: We can also create an
  inequality in the slope-intercept form from
  any two points and the words: less than (<),
  less than or equal to (≤), greater than(>),
  greater than and equal to(≥) accordingly.
  EX:
  Write the slope-intercept form of the line
  that is greater than or equal to and
  inequality that passes through the points
  (0, -0.5) and(2, -5.5)
  Use the given points and equation of slope:
  A(0,-0.5) B(2,-5.5)
  −
  −
  =
  −.−−.
  −
  =
  −
  
  = -
  
  We now use the slope and a point to find the
  y intercept (b).
  y ≥ mx + b
  -3 = Isolate b:
  
  -3 + = b
  
  b=- +
  
  () + b
  =-
  
  Going back to the equation:
  y = mx + b
  we replace what we have found:
  
  m = - and b = - ½
  To get the final slope-intercept form of the
  line passing through (3, -2) and(1, -3)
  y≥
  
  −
  
  x–½
  We now proceed to graph the equation:
  
  y≥- x-
  
  Y-intercept
  y crossing
  YOU TRY IT:
  Write the equation of the inequality.
  Use the given points and equation of slope:
  B(1,-3)
  A(3,-2)
  −
  −
  =
  −−−
  −
  =
  −
  −
  =
  
  We now use the slope and a point to find the
  y intercept (b).
  y < mx + b
  -3 =
  Isolate b:
  
  () + b
  -3 - = b
  b=-
  
  − =-
  
  Going back to the equation:
  y = mx + b
  we replace what we have found:
  m=
  
  and b = -
  
  To get the final slope-intercept form of the
  line passing through (3, -2) and(1, -3)
  y<
  
  x -3.5
  We now proceed to graph the equation:
  y<
  
  2
  1
  x-
  
  Y-intercept
  y crossing
  Real-World:
  A fish market charges \$9 per
  pound for cod and \$12 per pound
  per flounder. Let x = pounds of cod
  and y = pounds of flounder. What
  is the inequality that shows how
  much of each type of fish the store
  must sell per day to reach a daily
  quota of at least \$120?
  Real-World(SOLUTION):
  A fish market charges \$9 per pound for cod
  and \$12 per pound per flounder. Let x = pounds of
  cod and y = pounds of flounder. What is the
  inequality that shows how much of each type of
  fish the store must sell per day to reach a daily
  quota of at least \$120?
  Cod
  x
  Flounder  y
  At least \$120  9x + 12y ≥ 120
  SOLUTION:
  y≥-
  9x + 12y ≥ 120 
  
  x
  
  + 10
  10
  9
  8
  10
  8
  4
  4
  8
  10
  12
  Any point in the line or in the shaded
  region is a solution.
  YOU TRY IT:
  A music store sells used CDs
  for \$5 and buys used CDs for \$1.50.
  You go to the store with \$20 and
  some CDs to sell. You want to have
  at least \$10 left when you leave
  the store. Write and graph an
  inequality to show how many CDs
  Real-World(SOLUTION):
  A music store sells used CDs for \$5 and
  buys used CDs for \$1.50. You go to the store
  with \$20 and some CDs to sell. You want to
  have at least \$10 left when you leave the
  store. Write and graph an inequality to
  show how many CDs you could buy and sell.
  Bought CDs
  Sold CDs
   -5x
   +1.5y
  At least \$10 left  -5x + 1.5y ≥ -10
  NOTE: -10 since you spent this money.
  SOLUTION:
  -5x + 1.5y ≥ -10 
  y≥
  
  x
  .
  – 6.6
  10
  8
  6
  4
  2
  1
  2
  3
  4
  Any point in the line or in the shaded
  region is a solution.
  VIDEOS:
  Linear Inequalities
explanation: The document is a detailed explanation of graphing linear inequalities, including the slope-intercept form, how to determine the equation of an inequality from a graph, and several examples with solutions. It also includes real-world applications and practice problems with solutions. The document contains multiple questions and their corresponding answers.
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distilabel_metadata.content: 6_5 Graphing Linear Inequalities
  
  ```6.5 Slope intercept form
  for Inequalities:
  Linear Inequality: is a linear equation with
  an inequality sign (< , ≤, >, ≥)
  Solution of an Inequality: is an ordered
  pair (x, y) that makes the inequality true.
  GOAL:
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  Slope-Intercept Form: The linear equation of a
  nonvertical line with an inequality sign:
  y (<, ≤, >, ≥) m x + b
  Slope =
  −
  −
  =
  ⍙
  =
  ⍙
  y-intercept
  y crossing
  http://mathgraph.idwvogt.com/examples.html
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y<mx+b
  dash line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y>mx+b
  dash line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y≤mx+b
  Solid line
  Whenever we are given a graph we must be able
  to provide the equation of the function.
  y≥mx+b
  Solid line
  EX:
  Provide the equation of the inequality.
  Solution: Since line is dashed and shaded at
  the bottom we use <. Also, the inequality
  must be in
  slope-intercept form:
  Y < mx + b
  1. Find the y-intercept
  In this graph b = +1.
  2. Find another point
  to get the slope.
  A(0,5)
  A(0,1)
  B(3,-2)
  Use the equation of slope to find the slope:
  −
  −
  =
  −−
  −
  =
  −
  
  = -1
  The slope-intercept
  form inequality is:
  A(0,1)
  B(3,-2)
  y < -1x + 1
  Remember:
  This means that if you start a 1 and move down
  one and over to the right one, and continue this
  pattern. We shade the bottom since it is <.
  When work does not need to be shown:
  (EOC Test)
  look at the triangle
  points.
  Count the number
  of square going up
  or down and to the
  right.
  In this case 1 down
  and 1 right. Thus slope is -1/1 = -1
  YOU TRY IT
  Provide the equation of the
  inequality.
  YOU TRY IT: (Solution)
  The inequality is solid and shaded below:
  Y ≤ mx + b
  1. Find the y-intercept
  In this graph b = + 4.
  2. Find another point
  to get the slope.
  A(0,4) B(1,0)
  A(0,4)
  B(1,0)
  Use the equation of slope to find the slope:
  −
  −
  =
  −
  −
  =
  −
  
  =-4
  The slope-intercept
  form equation is:
  A(0,4)
  B(1,0)
  y ≤ -4x + 4
  Remember:
  This means that if you start a 4 and move down
  four and one over to the right. Solid line and
  shaded down means we must use ≤.
  When no work is required, you can use the
  rise/run of a right triangle between the two
  points:
  Look at the triangle,
  down 4 (-4) over to the
  right 1 (+1)
  slope = -4/+1 = -4
  A(0,4)
  B(1,0)
  Remember:
  You MUST KNOW BOTH procedures, the
  slope formula and the triangle.
  Given Two Points: We can also create an
  inequality in the slope-intercept form from
  any two points and the words: less than (<),
  less than or equal to (≤), greater than(>),
  greater than and equal to(≥) accordingly.
  EX:
  Write the slope-intercept form of the line
  that is greater than or equal to and
  inequality that passes through the points
  (0, -0.5) and(2, -5.5)
  Use the given points and equation of slope:
  A(0,-0.5) B(2,-5.5)
  −
  −
  =
  −.−−.
  −
  =
  −
  
  = -
  
  We now use the slope and a point to find the
  y intercept (b).
  y ≥ mx + b
  -3 = Isolate b:
  
  -3 + = b
  
  b=- +
  
  () + b
  =-
  
  Going back to the equation:
  y = mx + b
  we replace what we have found:
  
  m = - and b = - ½
  To get the final slope-intercept form of the
  line passing through (3, -2) and(1, -3)
  y≥
  
  −
  
  x–½
  We now proceed to graph the equation:
  
  y≥- x-
  
  Y-intercept
  y crossing
  YOU TRY IT:
  Write the equation of the inequality.
  Use the given points and equation of slope:
  B(1,-3)
  A(3,-2)
  −
  −
  =
  −−−
  −
  =
  −
  −
  =
  
  We now use the slope and a point to find the
  y intercept (b).
  y < mx + b
  -3 =
  Isolate b:
  
  () + b
  -3 - = b
  b=-
  
  − =-
  
  Going back to the equation:
  y = mx + b
  we replace what we have found:
  m=
  
  and b = -
  
  To get the final slope-intercept form of the
  line passing through (3, -2) and(1, -3)
  y<
  
  x -3.5
  We now proceed to graph the equation:
  y<
  
  2
  1
  x-
  
  Y-intercept
  y crossing
  Real-World:
  A fish market charges \$9 per
  pound for cod and \$12 per pound
  per flounder. Let x = pounds of cod
  and y = pounds of flounder. What
  is the inequality that shows how
  much of each type of fish the store
  must sell per day to reach a daily
  quota of at least \$120?
  Real-World(SOLUTION):
  A fish market charges \$9 per pound for cod
  and \$12 per pound per flounder. Let x = pounds of
  cod and y = pounds of flounder. What is the
  inequality that shows how much of each type of
  fish the store must sell per day to reach a daily
  quota of at least \$120?
  Cod
  x
  Flounder  y
  At least \$120  9x + 12y ≥ 120
  SOLUTION:
  y≥-
  9x + 12y ≥ 120 
  
  x
  
  + 10
  10
  9
  8
  10
  8
  4
  4
  8
  10
  12
  Any point in the line or in the shaded
  region is a solution.
  YOU TRY IT:
  A music store sells used CDs
  for \$5 and buys used CDs for \$1.50.
  You go to the store with \$20 and
  some CDs to sell. You want to have
  at least \$10 left when you leave
  the store. Write and graph an
  inequality to show how many CDs
  Real-World(SOLUTION):
  A music store sells used CDs for \$5 and
  buys used CDs for \$1.50. You go to the store
  with \$20 and some CDs to sell. You want to
  have at least \$10 left when you leave the
  store. Write and graph an inequality to
  show how many CDs you could buy and sell.
  Bought CDs
  Sold CDs
   -5x
   +1.5y
  At least \$10 left  -5x + 1.5y ≥ -10
  NOTE: -10 since you spent this money.
  SOLUTION:
  -5x + 1.5y ≥ -10 
  y≥
  
  x
  .
  – 6.6
  10
  8
  6
  4
  2
  1
  2
  3
  4
  Any point in the line or in the shaded
  region is a solution.
  VIDEOS:
  Linear Inequalities
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed explanation of graphing linear inequalities, including the slope-intercept form, how to determine the equation of an inequality from a graph, and several examples with solutions. It also includes real-world applications and practice problems with solutions. The document contains multiple questions and their corresponding answers.
  
  Label: multiple-question-and-answer