Example Variable: Let x represent the number. Inequality: 2/3x+6 ≥ 22 Solution Set: {x | x≥24 }
Practice makes perfect
To algebraically express a verbal inequality, we will need to translate the given information into mathematical symbols and operations.
Writing the Inequality
The phrase is at least can be expressed as ≥. This symbol will be in the middle of our expression.d
... ≥...
On the left-hand side of the inequality symbol, we will translate any verbal expression that comes beforeis at least. If we let x represent a number, we can form an algebraic expression. Note that x is an example variable — we can use any letter to represent it.
Two thirds of a number added to six
2/3 x + 6
On the right-hand side of the inequality symbol, we will translate any verbal expression that comes afteris at least.
twenty-two
22
Finally, we can bring these two expressions together to form the inequality.
2/3x+6≥ 22
This solution tells us that all values greater than or equal to 24 satisfy the inequality. Let's write the solution set.
{x | x≥ 24}
Checking Our Solution
We can check our solution by substituting a few arbitrary values into the inequality translated in the previous part. The value satisfies the inequality if the inequality remains true after substituting and simplifying.
x
2/3x+6 ≥ 22
Evaluate
21
2/3 ( 21)+6 ? ≥ 22
20 ≱ 22
24
2/3( 24)+6 ? ≥ 22
22 ≥ 22
27
2/3( 27)+6 ? ≥ 22
24≥ 22
We can conclude that, as long as x is greater than or equal to 24, the inequality is satisfied. This means the solution set we have found is correct.
