Algebra 1 - Linear Inequalities

Introduction:

Mr. Crawford is ordering pizzas and breadsticks for a school pizza party and has a budget of \($81\), but no more.   An order of breadsticks costs \($7\), and a pepperoni pizza cost \($13\).

The budget causes a constraint that needs to be considered.   The phrase "no more" means Mr. Crawford can spend \($81\) or less.   This leads to an inequality.   Let \(x\) denote the number of orders of breadsticks and \(y\) be the number of pepperoni pizzas.   Then the inequality looks like:   \(7x+13y\le81\).

One possible combination is \(2\) orders of breadsticks and \(4\) pepperoni pizzas:
\(\implies7(2)+13(4)=14+52\implies66\le81\)   TRUE.

Another combination is \(4\) orders of breadsticks and \(3\) pepperoni pizzas:
\(\implies7(4)+13(3)=28+39\implies67\le81\)   TRUE.

The combination of \(6\) orders of breadsticks and \(3\) pepperoni pizzas gives exactly \($81\):
\(\implies7(6)+13(3)=42+39\implies81\le81\)   TRUE.

However, the combination of \(6\) orders of breadsticks and \(5\) pepperoni pizzas does NOT work:
\(\implies7(6)+13(5)=42+65\implies107\nleq81\)   FALSE.

breadsticks pizzas

The graph shows the line \(7x+13y=81\), and the combinations that gave TRUE are either ON or BELOW the line.   The combination that gave FALSE is ABOVE the line.   So how can ALL possible combinations be found?   There are two things to consider to find all possible solutions:

  1. Are points ON the line solutions?
  2. Are the points ABOVE or BELOW the line solutions?

The first question is completely determined by the inequality symbol used in the problem.   The following table summarizes:

Symbol Points ON the line... Line is...
\(<\)   or   \(>\) are NOT solutions a dotted line
\(\le\)   or   \(\ge\) are solutions a solid line

In the breadsticks and pepperoni pizzas examples above, the symbol from the inequality is \(\le\), so points ON the line are solutions and the graph shows a solid line.

For the second question, pick a point and test for TRUE or FALSE in the inequality.

  • If the test gives TRUE, then shade the side of the line the point is located in.
  • If the test gives FALSE, then shade the other side of the line.

In the example above with breadsticks and pepperoni pizzas, the points below the line tested TRUE, while the points above the line tested FALSE.   So the shading is below the line.

This shaded region shows ALL the possible solutions to the inequality.

breadsticks pizzas #2

Video Example #1


Video Example #2


Video Example #3