Example Variable: Let x represent the number. Inequality: 34x-9≥ 42 Solution Set: {x| x≥ 68}
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 at the center of our expression.
... ≥...
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.
three fourths of a number decreased by nine
3/4 x -9
On the right-hand side of the inequality symbol, we will translate any verbal expression that comes afteris at least.
forty-two
42
Finally, we can bring these two expressions together to form the inequality.
3/4x-9≥ 42
This solution tells us that all values greater than or equal to 68 satisfy the inequality. Let's write the solution set.
{x|x≥ 68}
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
3/4x-9≥ 42
Evaluate
64
3/4( 64)-9≥ 42
39 ≱ 42
68
3/4( 68)-9≥ 42
42 ≥ 42
72
3/4( 72)-9≥ 42
45 ≥ 42
We can conclude that, as long as x is greater than or equal to 68, the inequality is satisfied. This means the solution set we have found is correct.
