2.7 Linear inequalities and absolute value inequalities  (Page 4/11)

We want the distance between $x$ and 5 to be less than or equal to 4. We can draw a number line, such as in [link] , to represent the condition to be satisfied.

The distance from $x$ to 5 can be represented using an absolute value symbol, $\left|x-5\right|.$ Write the values of $x$ that satisfy the condition as an absolute value inequality.

We need to write two inequalities as there are always two solutions to an absolute value equation.

If the solution set is $x\le 9$ and $x\ge 1,$ then the solution set is an interval including all real numbers between and including 1 and 9.

So $\left|x-5\right|\le 4$ is equivalent to $\left[1,9\right]$ in interval notation.

We are trying to determine where $y<0,$ which is when $-\frac{1}{2}|4x-5|+3<0.$ We begin by isolating the absolute value.

Next, we solve for the equality $|4x-5|=6.$

Now, we can examine the graph to observe where the y- values are negative. We observe where the branches are below the x- axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $x=-\frac{1}{4}$ and $x=\frac{11}{4},$ and that the graph opens downward. See [link] .