2.7: Solve Linear Inequalities

Learning Objectives
Note

Before you get started, take this readiness quiz.

  1. Translate from algebra to English: \(15>x\).
    If you missed this problem, review Exercise 1.3.1.
  2. Solve: \(n−9=−42\).
    If you missed this problem, review Exercise 2.1.7.
  3. Solve: \(−5p=−23\).
    If you missed this problem, review Exercise 2.2.1.
  4. Solve: \(3a−12=7a−20\).
    If you missed this problem, review Exercise 2.3.22.

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality \(x > 3\) true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality \(x > 3\).

We show the solutions to the inequality \(x > 3\) on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of \(x > 3\) is shown in Figure \(\PageIndex\). Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater <a href=than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis." />

The graph of the inequality \(x \geq 3\) is very much like the graph of \(x > 3\), but now we need to show that 3 is a solution, too. We do that by putting a bracket at \(x = 3\), as shown in Figure \(\PageIndex\).

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on <a href=the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket." />

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

Exercise \(\PageIndex\)

Graph on the number line:

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on <a href=the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket." />

1. \(x\leq 1\) This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at x=1 to show that it is included.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 5 is graphed on <a href=the number line, with an open parenthesis at x equals 5, and a red line extending to the right of the parenthesis." />

2. \(x

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 1 is graphed on <a href=the number line, with an open parenthesis at x equals negative 1, and a red line extending to the right of the parenthesis." />

3. \(x>−1\) This means all numbers greater than −1, but not including −1. We shade in all the numbers on the number line to the right of −1, then put a parenthesis at x=−1 to show it is not included.

Exercise \(\PageIndex\)

Graph on the number line:

  1. \(x\leq −1\)
  2. \(x>2\)
  3. \(x
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1 is graphed on <a href=the number line, with an open bracket at x equals negative 1, and a dark line extending to the left of the bracket." />
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on <a href=the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis." />
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less <a href=than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the left of the parenthesis." />

Exercise \(\PageIndex\)

Graph on the number line:

  1. \(x>−2\)
  2. \(x
  3. \(x\geq −1\)
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 2 is graphed on <a href=the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the right of the parenthesis." />
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on <a href=the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis." />
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on <a href=the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket." />

We can also represent inequalities using interval notation. As we saw above, the inequality \(x>3\) means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express \(x>3\) as \((3, \infty)\). The symbol \(\infty\) is read as ‘infinity’. It is not an actual number. Figure \(\PageIndex\) shows both the number line and the interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater <a href=than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis." />

The inequality \(x\leq 1\) means all numbers less than or equal to 1. There is no lower end to those numbers. We write \(x\leq 1\) in interval notation as \((-\infty, 1]\). The symbol \(-\infty\) is read as ‘negative infinity’. Figure \(\PageIndex\) shows both the number line and interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on <a href=the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket." />

INEQUALITIES, NUMBER LINES, AND INTERVAL NOTATION

This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to <a href=the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket." />

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure \(\PageIndex\).

This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on <a href=the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”" />

Exercise \(\PageIndex\)

Graph on the number line and write in interval notation.

.
Shade to the right of −3, and put a bracket at −3. .
Write in interval notation. .
the left of 2.5, and put a parenthesis at 2.5.” To the right of this sentence is a number line ranging from 0 to 3, with tick marks at each integer. There is a parenthesis at 2.5 (written in) and a red line extends to the left from 2.5. Another line down on the left, the instructions say: “Write in interval notation.” To the right of this sentence is the interval notation: parenthesis, negative infinity comma 2.5, parethesis.">
.
Shade to the left of 2.5, and put a parenthesis at 2.5. .
Write in interval notation. .
the left of negative 3/5, and put a bracket at negative 3/5.” To the right of this sentence is a number line ranging from negative 2 to 1, with tick marks at each integer. There is a bracket at negative 3/5 (written in) and a red line extends to the left from negative 3/5. Another line down on the left, the instructions say: “Write in interval notation.” To the right of this sentence is the interval notation: parenthesis, negative infinity comma negative three fifths, bracket.">
.
Shade to the left of \(-\frac\), and put a bracket at \(-\frac\). .
Write in interval notation. .
Exercise \(\PageIndex\)

Graph on the number line and write in interval notation:

  1. \(x>2\)
  2. \(x\leq −1.5\)
  3. \(x\geq \frac\)
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on <a href=the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 2 comma infinity, parenthesis." />
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on <a href=the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 1.5, bracket." />
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3/4 is graphed on <a href=the number line, with an open bracket at x equals 3/4, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 3/4 comma infinity, parenthesis." />

Exercise \(\PageIndex\)

Graph on the number line and write in interval notation:

  1. \(x\leq −4\)
  2. \(x\geq 0.5\)
  3. \(x\)
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 4 is graphed on <a href=the number line, with an open bracket at x equals negative 4, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 4, bracket." />
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 0.5 is graphed on <a href=the number line, with an open bracket at x equals 0.5, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, o.5 comma infinity, parenthesis." />
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2/3 is graphed on <a href=the number line, with an open parenthesis at x equals negative 2/3, and a dark line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 2/3, parenthesis." />

Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.