The phrase is no more than can be expressed as ≤.
Example Variable: Let x represent the number. Inequality: 10 ≤ 4(2x+3) Solution Set: {x|x ≥ - 14}
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 no more than can be expressed as ≤. This symbol will be in the middle of our expression.
... ≤...
On the left-hand side of the inequality symbol, we will translate any verbal expression that comes beforeis no more than.
ten
10
On the right-hand side of the inequality symbol, we will translate any verbal expression that comes afteris no more than. 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.
4times the sum of twice a number and three
4 ( 2 x + 3 )
Finally, we can bring these two expressions together to form the inequality.
10 ≤ 4(2x+3)
This tells us that all values greater than or equal to - 14 satisfy the inequality. Let's write the solution set.
{x|x≥ -1/4}
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
10 ≤ 4 (2x+3)
Evaluate
-1/2
10 ? ≤ 4(2( -1/2)+3)
10 ≰ 8
-1/4
10 ? ≤ 4(2( - 1/4)+3)
10 ≤ 10
-1/8
10 ? ≤ 4(2( - 1/8)+3)
10 ≤ 11
We can conclude that, as long as x is greater than or equal to - 14, the inequality is satisfied. This means the solution set we have found is correct.
