In this section you will learn to:

• Solve compound inequalities—OR
  • Solve compound inequalities in the form of or and express the solution graphically and with an interval
• Solve compound inequalities—AND
  • Express solutions to inequalities graphically and with interval notation
  • Identify solutions for compound inequalities in the form [latex]a

Disjunctions: Solve compound inequalities in the form of or and express the solution graphically and in interval notation

As we saw in the last section, the solution of a compound inequality that consists of two inequalities joined with the word or is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.

In this section you will see that some inequalities need to be simplified before their solution can be written or graphed.

In the following example, you will see an example of how to solve a one-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.

Remember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.

In the last example, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution “all real numbers.”  Any real number will produce a true statement for either [latex]y<3\text{ or }y\ge -4[/latex] when it is substituted for x.

The following video contains an example of solving a compound inequality involving OR, and drawing the associated graph.

In the next section you will see examples of how to solve compound inequalities containing and.

Conjunctions: Solve compound inequalities in the form of and and express the solution graphically and in interval notation

The solution of a compound inequality that consists of two inequalities joined with the word and is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an and compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this is where the two graphs overlap.

In this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.

Tripartite Inequalities: Compound inequalities in the form [latex]a

Compound inequalities in the form [latex]aand.  With these problems, you can split a compound inequality in the form of  [latex]a and [latex]x>a[/latex], or you can solve the inequality more quickly by applying the properties of inequality to all three segments of the compound inequality.

In the example below, we will show how to apply the properties of inequality to all three segments of the compound inequality. In the video below the example, we will show how to split it into two inequalities to solve. Read and view both examples to see if you prefer one method over the other.

Example 6

Solve for x: [latex]\hspace{.05in} 3\lt2x+3\leq 7[/latex]

Show Solution

 

Isolate the variable by subtracting 3 from all 3 parts of the inequality, then dividing each part by 2.

[latex]\begin{array}{r}\,\,\,\,3\,\,\lt\,\,2x+3\,\,\leq \,\,\,\,7\\\underline{\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\underline{\,-3}\,\\\,\,\,\,\,\underline{\,0\,}\,\,\lt\,\,\,\,\underline{2x}\,\,\,\,\,\,\,\,\leq\,\,\,\underline{\,4\,}\\2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,0\lt x\leq 2\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Answer

Inequality: [latex]\displaystyle 0\lt{x}\le 2[/latex]

Interval: [latex]\left(0,2\right][/latex]

Graph:

The following video works through two examples of solving tripartite inequalities.

 

Alternately, a tripartite inequality can be solved by formally breaking it into an and compound inequality, as shown in the video below.

To solve inequalities of the form [latex]aand compound inequality or use the addition and multiplication properties of inequality to solve the tripartite inequality for x directly. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.

The solution to a compound inequality with and is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word and:

Possible Cases for Conjunction
Case 1:
Description	The solution could be all the values between two endpoints
Example of Inequalities	[latex]x\le{1}[/latex] and [latex]x\gt{-1}[/latex], or as a bounded inequality: [latex]{-1}\lt{x}\le{1}[/latex]
Initial Intervals	[latex]\left(-\infty,1\right][/latex] and [latex]\left(-1,\infty\right)[/latex]
Graphs	The intersection is the overlap between the two regions, which will include only values between [latex]-1[/latex] and [latex]1[/latex] (not including [latex]-1[/latex] itself).
Final Answer in Interval Notation	[latex](-1,1][/latex]
Case 2:
Description	The solution could begin at a point on the number line and extend in one direction.
Example of Inequalities	[latex]x\gt -3[/latex] and [latex]x\ge4[/latex]
Initial Intervals	[latex](-3,\infty)[/latex] and  [latex]\left[4,\infty\right)[/latex]
Graphs	The two shaded regions overlap for all values greater than or equal to [latex]4[/latex].
Final Answer in Interval Notation	[latex][4,\infty)[/latex]
Case 3:
Description	There is no solution to the compound inequality
Example of Inequalities	[latex]x\lt{-3}[/latex] and [latex]x\gt{3}[/latex]
Intervals	[latex]\left(-\infty,-3\right)[/latex] and [latex]\left(3,\infty\right)[/latex]
Graph	Since there is no overlap, there is no solution to highlight on the graph as shown below.
Final Answer in Interval Notation	No Solution (equivalently, the empty set [latex]\{ \hspace{.05in} \}[/latex] or [latex]\varnothing[/latex])

In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.

Example 7

Solve for x: [latex]\hspace{.05in} x+2>5[/latex] and [latex]x+4<5[/latex]

Show Solution

Solve each inequality separately.

[latex]\displaystyle \begin{array}{l}x+2>5\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,x+4<5\,\,\,\,\\\underline{\,\,\,\,\,-2\,-2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-4\,-4}\\x\,\,\,\,\,\,\,\,>\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,<\,1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,x>3\,\,\,\,\text{and}\,\,\,\,x<1\end{array}[/latex]

Find the overlap between the solutions.

Answer

There is no overlap between [latex]\displaystyle x>3[/latex] and [latex]x<1[/latex], so there is no solution. Since there is no overlap, there is no solution to highlight on the graph as shown below.