Concept Byte: Quadratic Inequalities
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What does the absolute value of a real number represent? What does an absolute value inequality imply?
See solution.
There are many ways to solve the given inequality. We will discuss how to do this algebraically and graphically.
Recall that the absolute value of a real number, a, expressed as |a|, is the distance between a and 0 on the number line. Therefore, the inequality |- 3x+4|>0 will be true for all values of x for which -3 x+4 ≠ 0. We can solve for the value of x for which -3x+4 = 0, and exclude that value to find the solution set.
Now, we want to use graph the function y =|-3x+4| to solve the given inequality. We will use a table of values. When using this method, we need to choose appropriate values of x. We will take the x-coordinate of the vertex and two other values, one to the left and one to the right of the vertex.
The x-coordinate of the vertex of an absolute value equation is always the value of x which makes the expression inside the absolute value equal to 0. Let's take a look at the expression inside the absolute value for the given equation. -3 x+4 = 0 ⇔ x = 4/3
Now we can make the table of values.
| x | y=|-3 x+4| | Simplify | y |
|---|---|---|---|
| 0 | y=|-3 ( 0)+4| | |0+4| | 4 |
| 4/3 | y=|-3 ( 4/3)+4| | |- 4+4| | 0 |
| 2 | y=|-3 ( 2)+4| | |-6 + 4| | 2 |
We can plot these ordered pairs on a coordinate plane and connect them to get the graph of the equation.
We can see that all the x-values except for x= 43 make the function y=|-3x+4| greater than 0, satisfying the inequality |-3 x+4|>0.