Solve Compound Inequalities in the Form of “and” and Express the Solution Graphically

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 previous 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.

Compound Inequalities in the Form [latex]a

Rather than splitting a compound inequality in the form of  [latex]a and [latex]x>a[/latex], you can more quickly solve the inequality by applying the properties of inequalities to all three segments of the compound inequality.

Example

Solve for x. [latex]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]

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

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

Graph:

In the video below, you will see another example of how to solve an inequality in the form  [latex]a

Example

Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[/latex].

Show Solution

Let us try the first method: write two inequalities.

[latex]\begin{array}{lll}3+x> 7x - 2\hfill & \textit{and}\hfill & 7x - 2> 5x - 10\hfill \\ 3> 6x - 2\hfill & \hfill & 2x - 2> -10\hfill \\ 5> 6x\hfill & \hfill & 2x> -8\hfill \\ \dfrac{5}{6}> x\hfill & \hfill & x> -4\hfill \\ x< \dfrac{5}{6}\hfill & \hfill & -4< x\hfill \end{array}[/latex]

The solution set is [latex]-4

To solve inequalities like [latex]ax. 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:

Case 1:
Description	The solution could be all the values between two endpoints
Inequalities	[latex]x\le{1}[/latex] and [latex]x\gt{-1}[/latex], or as a bounded inequality: [latex]{-1}\lt{x}\le{1}[/latex]
Interval	[latex]\left(-1,1\right][/latex]
Graphs
Case 2:
Description	The solution could begin at a point on the number line and extend in one direction.
Inequalities	[latex]x\gt3[/latex] and [latex]x\ge4[/latex]
Interval	[latex]\left[4,\infty\right)[/latex]
Graphs
Case 3:
Description	In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality
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

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

Example

Solve for x. [latex]x+2>5[/latex] and [latex]x+4<5[/latex]

Show Solution

Solve each inequality separately.

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

Find the overlap between the solutions.

There is no overlap between [latex]\displaystyle x>3[/latex] and [latex]x<1[/latex], so there is no solution.

Summary

A compound inequality is a statement of two inequality statements linked together either by the word or or by the word and. Sometimes, an and compound inequality is shown symbolically, like [latex]aand. Because compound inequalities represent either a union or intersection of the individual inequalities, graphing them on a number line can be a helpful way to see or check a solution. Compound inequalities can be manipulated and solved in much the same way any inequality is solved, by paying attention to the properties of inequalities and the rules for solving them.