Solving Inequalities
Common Core Standard: Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.
Common Core Standard: Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q,
and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
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Greater than >, this means on a number line the shading would be any number bigger than this number
including rational numbers (fractions and decimals)
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Less than <, this means on a number line that shading would be any number smaller than this number
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Greater than or equal to ≥, this means on a number line that shading would be any number greater than this
number and would include shading in the circle above the number, because the variable could equal that
number
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Less than or equal to ≤, this means on a number line that shading would be any number less than this number
and would include shading in the circle above the number, because the variable could also be that number
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Why number lines?
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Since these are inequalities, it means that the variable could be a range of numbers
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The number line shows the all the possible numbers the variable could be
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This is why the shading on the number line is the answer, and this is also why the arrow on the number
line must be shaded to show that any number that is either greater or less regardless of if it’s written
on the number line
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Number line example: 6x + 4 > 16
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Like in solving equations, first start by combining like terms
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The like terms are 4 and 16
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To get the 4 with the 16 subtract 4 from both sides so that it equals zero
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6x + 4 – 4 > 16 – 4
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6x > 12
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Now, divide both sides by 6 so that the variable is 1x instead of 6x
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x > 2, Note: the line above the 2 on the number line represents the shading
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Real World Example:
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In high school, Jordan gets a job where he earns $40 per week plus $5 per sale. One week Jordan wants
his pay to be at least $100. Write an inequality for the number of sales Jordan needs to make, and
describe the solutions.
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1st: Determine what is known in the inequality and what is not known, in this case, we do not know how
many sales Jordan needs to make this is the variable
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n = number of sales
2nd: Determine what type of inequality this is (greater than, less that, greater than or equal to, etc.)
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This is greater than or equal to, because Jordan wants to earn at least $100, which means he
could earn exactly $100 or more than $100
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3rd: Determine how the variable and the other terms are related
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The $100 is what is going to be greater than or equal to
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He will already earn $40 regardless of his sales
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The sales are $5 per sale this is dependent on the number of sales, which is the variable; so $5
is being multiplied by the variable n
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This $5n is added to $40, since he already earns this
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Inequality: $5n + $40 ≥ $100
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Now, solve like the first example
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Combine like terms of $40 and $100
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$5n + $40 - $40 ≥ $100 - $40
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$5n ≥ $60
Divide both sides of the inequality by $5
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$5n ÷ $5 ≥ $60 ÷ $5
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n ≥ 12 sales
This means that Jordan must at least make 12 sales in 1 week to earn $100 for that week.
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Special Circumstances: When solving inequalities, if when getting the variable to equal 1x or 1 times the
variable, you need to multiply or divide by a negative number, the inequality symbol flips
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For instance, a > becomes a < and a ≥ becomes ≤
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Note: When the inequality symbol flips, if it was already an equal to it remains equal to
Example with negatives: -4h + 2 ≤ 18
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1st: combine like terms, 2 and 18
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-4h + 2 – 2 ≤ 18 – 2
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-4h ≤ 16
2nd: Divide by -4, so that the variable equals 1h
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-4h ÷ -4 ≤ 16 ÷ -4
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Since this had to be divided by a negative the inequality symbol flips to become ≥
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So h ≥ -4
Proof: if h is any number bigger or equal to -4, it could be 0
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Substitute 0 into the original inequality -4h + 2 ≤ 18
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-4 ● 0 + 2 ≤ 18
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2 ≤ 18, this is true, since 2 is less than or equal to 18