The first method is to write two separate inequalities: [latex]3\le 2x+2[/latex] and [latex]2x+2<6[/latex]. We solve them independently.

Then, we can rewrite the solution as a compound inequality, the same way the problem began.

In interval notation, the solution is written as [latex]\left[\frac{1}{2},2\right)[/latex].

The second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time.

We get the same solution: [latex]\left[\frac{1}{2},2\right)[/latex].

Let’s try the first method. Write two inequalities:

The solution set is [latex]-4<x<\frac{5}{6}[/latex] or in interval notation [latex]\left(-4,\frac{5}{6}\right)[/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.

We want the distance between [latex]x[/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied.

The distance from [latex]x[/latex] to 5 can be represented using an absolute value symbol, [latex]|x - 5|[/latex]. Write the values of [latex]x[/latex] 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 [latex]x\le 9[/latex] and [latex]x\ge 1[/latex], then the solution set is an interval including all real numbers between and including 1 and 9.

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

[latex]\begin{array}{c}|x - 1|\le 3\hfill \\ \hfill \\ -3\le x - 1\le 3\hfill \\ \hfill \\ -2\le x\le 4\hfill \\ \hfill \\ \left[-2,4\right]\hfill \end{array}[/latex]

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

Next, we solve [latex]|4x - 5|=6[/latex].

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 [latex]x=-\frac{1}{4}[/latex] and [latex]x=\frac{11}{4}[/latex] and that the graph opens downward.