A linear inequality 138 is a mathematical statement that relates a linear expression as either less than or greater than another. The following are some examples of linear inequalities, all of which are solved in this section:
| \(5 x + 7 < 22\) | \(- 2 ( x + 8 ) + 6 \geq 20\) | \(- 2 ( 4 x - 5 ) < 9 - 2 ( x - 2 )\) |
A solution to a linear inequality 139 is a real number that will produce a true statement when substituted for the variable. Linear inequalities have either infinitely many solutions or no solution. If there are infinitely many solutions, graph the solution set on a number line and/or express the solution using interval notation.
Are \(x=−4\) and \(x=6\) solutions to \(5x+7
Solution
Substitute the values in for \(x\), simplify, and check to see if we obtain a true statement.
Check \(x=−4\)
Check \(x=6\)
Answer:
\(x=−4\) is a solution and \(x=6\) is not
All but one of the techniques learned for solving linear equations apply to solving linear inequalities. You may add or subtract any real number to both sides of an inequality, and you may multiply or divide both sides by any positive real number to create equivalent inequalities. For example:
Subtracting \(7\) from each side and dividing each side by positive \(5\) results in an inequality that is true.
Solve and graph the solution set: \(5x+7
Solution
It is helpful to take a minute and choose a few values in and out of the solution set, substitute them into the original inequality, and then verify the results. As indicated, you should expect \(x=0\) to solve the original inequality and that \(x=5\) should not.
Check \(x=0\)
Check \(x=5\)
Checking in this manner gives us a good indication that we have solved the inequality correctly.
We can express this solution in two ways: using set notation and interval notation.
In this text we will choose to present answers using interval notation.
Answer
When working with linear inequalities, a different rule applies when multiplying or dividing by a negative number. To illustrate the problem, consider the true statement \(10 > −5\) and divide both sides by \(−5\).
Dividing by \(−5\) results in a false statement. To retain a true statement, the inequality must be reversed.
The same problem occurs when multiplying by a negative number. This leads to the following new rule: when multiplying or dividing by a negative number, reverse the inequality. It is easy to forget to do this so take special care to watch for negative coefficients. In general, given algebraic expressions \(A\) and \(B\), where \(c\) is a positive nonzero real number, we have the following properties of inequalities 140 :
We use these properties to obtain an equivalent inequality 141 , one with the same solution set, where the variable is isolated. The process is similar to solving linear equations.
Solve and graph the solution set: \(−2(x+8)+6≥20\).
Solution
Answer:
Interval notation \((−∞, −15] \)
Solve and graph the solution set: \(−2(4x−5)
Solution
Answer:
Interval notation \((−\frac, ∞)\)
Solve and graph the solution set: \(\fracx−2≥\frac(\fracx−9)+1\).
Solution
Answer:
Interval notation: \((−∞, 4]\)
Solve and graph the solution set: \(10 - 5 ( 2 x + 3 ) \leq 25\)
Answer
Following are some examples of compound linear inequalities:
| \(- 13 < 3 x - 7 < 17\) | \(4 x + 5 \leq - 15 \text < or >6 x - 11 > 7\) |
These compound inequalities 142 are actually two inequalities in one statement joined by the word and or by the word or. For example,
is a compound inequality because it can be decomposed as follows:
We can solve each inequality individually; the intersection of the two solution sets solves the original compound inequality. While this method works, there is another method that usually requires fewer steps. Apply the properties of this section to all three parts of the compound inequality with the goal of isolating the variable in the middle of the statement to determine the bounds of the solution set.
Solve and graph the solution set: \(−13
Solution
Answer:
Interval notation: \((−2,8)\)
Solve and graph the solution set: \(\frac≤\frac(\fracx+4)
Solution
Answer:
Interval notation \([−3,4)\)
It is important to note that when multiplying or dividing all three parts of a compound inequality by a negative number, you must reverse all of the inequalities in the statement. For example:
The answer above can be written in an equivalent form, where smaller numbers lie to the left and the larger numbers lie to the right, as they appear on a number line.
Use interval notation, write: \((-10, 5)\).
Solve and graph the solution set: \(−3≤−3(2x−3)
Answer
For compound inequalities with the word “or” you work both inequalities separately and then consider the union of the solution sets. Values in this union solve either inequality.
Solve and graph the solution set: \(4x+5≤−15\) or \(6x−11>7\).
Solution
Solve each inequality and form the union by combining the solution sets.
\(\begin 4 x + 5 & \leq - 15 \\ 4 x & \leq - 20 \\ x & \leq - 5 \end\)
Answer:
Interval notation \((−∞,−5]∪(3,∞)\)
Solve and graph the solution set: \(5 ( x - 3 ) < - 20 \text < or >2 ( 5 - 3 x ) < 1\).
Answer
\(( - \infty , - 1 ) \cup \left( \frac < 3 > < 2 >, \infty \right)\)
Some of the key words and phrases that indicate inequalities are summarized below:
| Key Phrases | Translation |
|---|---|
| A number is at least \(5\). | \(x\geq 5\) |
| A number is \(5\) or more inclusive. | \(x\geq 5\) |
| A number is at most \(3\). | \(x\leq 3\) |
| A number is \(3\) or less inclusive. | \(x\leq 3\) |
| A number is strictly less than \(4\). | \(x |
| A number is less than \(4\), noninclusive. | \(x |
| A number is greater than \(7\). | \(x>7\) |
| A number is more than \(7\), noninclusive. | \(x>7\) |
| A number is in between \(2\) and \(10\). | \(2 |
| A number is at least \(5\) and at most \(15\). | \(5\leq x\leq 15\) |
| A number may range from \(5\) to \(15\). | \(5\leq x\leq 15\) |
As with all applications, carefully read the problem several times and look for key words and phrases. Identify the unknowns and assign variables. Next, translate the wording into a mathematical inequality. Finally, use the properties you have learned to solve the inequality and express the solution graphically or in interval notation.
Seven less than \(3\) times the sum of a number and \(5\) is at most \(11\). Find all numbers that satisfy this condition.
Solution
First, choose a variable for the unknown number and identify the key words and phrases.
Let n represent the unknown indicated by “a number.”
\(\begin 3 ( n + 5 ) - 7 & \leq 11 \\ 3 n + 15 - 7 & \leq 11 \\ 3 n + 8 & \leq 11 \\ 3 n & \leq 3 \\ n & \leq 1 \end\)
Answer:
Any number less than or equal to \(1\) will satisfy the statement.
To earn a B in a mathematics course the test average must be at least \(80\)% and less than \(90\)%. If a student earned \(92\)%, \(96\)%, \(79\)%, and \(83\)% on the first four tests, what must she score on the fifth test to earn a B?
Solution
Set up a compound inequality where the test average is between \(80\)% and \(90\)%. In this case, include the lower bound, \(80\).
Let x represent the score on the fifth test.
Answer:
She must earn a score of at least \(50\)% and less than \(100\)%.
In the previous example, the upper bound \(100\)% was not part of the solution set. What would happen if she did earn a \(100\)% on the fifth test?
As we can see, her average would be \(90\)%, which would earn her an A.
Determine whether or not the given value is a solution.
Graph all solutions on a number line and provide the corresponding interval notation.
Figure 1.8.12
23. \(\left( - \infty , - \frac < 3 > < 2 >\right]\);
Graph all solutions on a number line and provide the corresponding interval notation.
Find all numbers that satisfy the given condition.
Set up an algebraic inequality and then solve.
1. Members may play \(4\) rounds or fewer.
3. Mark must earn at least \(9\) points on the fourth quiz.
5. He must score a \(15.1\) on the sixth event.
7. The angle is between \(90\) degrees and \(135\) degrees.
9. The computer will shut down when the temperature exceeds \(113\)°F.
1. Answer may vary
3. Answer may vary
138 Linear expressions related with the symbols \(≤, \).
139 A real number that produces a true statement when its value is substituted for the variable.
140 Properties used to obtain equivalent inequalities and used as a means to solve them.
141 Inequalities that share the same solution set.
142 Two or more inequalities in one statement joined by the word “and” or by the word “or.”
This page titled 1.8: Solving Linear Inequalities with One Variable is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.
