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2.7 Solve linear inequalities  (Page 2/8)

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The inequality x ≤ 1 means all numbers less than or equal to 1. There is no lower end to those numbers. We write x ≤ 1 in interval notation as ( − ∞ , 1 ] . The symbol − ∞ is read as ‘negative infinity’. [link] 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 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.
The inequality x ≤ 1 is graphed on this number line and written in interval notation.

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 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 [link] .

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 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.”
The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

Graph on the number line and write in interval notation.

ⓐ x ≥ −3 ⓑ x < 2.5 ⓒ x ≤ − 3 5

Solution

  1. ⓐ
    .
    Shade to the right of −3 , and put a bracket at −3 . .
    Write in interval notation. .

  2. ⓑ
    .
    Shade to the left of 2.5 , and put a parenthesis at 2.5 . .
    Write in interval notation. .

  3. ⓒ
    .
    Shade to the left of − 3 5 , and put a bracket at − 3 5 . .
    Write in interval notation. .
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Graph on the number line and write in interval notation:

ⓐ x > 2 ⓑ x ≤ − 1.5 ⓒ x ≥ 3 4

  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 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 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 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.
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Graph on the number line and write in interval notation:

ⓐ x ≤ − 4 ⓑ x ≥ 0.5 ⓒ x < − 2 3

  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 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 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 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.
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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.

Properties of equality

Subtraction Property of Equality Addition Property of Equality For any numbers a , b , and c , For any numbers a , b , and c , if a = b , then a − c = b − c . if a = b , then a + c = b + c .

Similar properties hold true for inequalities.

For example, we know that −4 is less than 2. .
If we subtract 5 from both quantities, is the
left side still less than the right side?		 .
We get −9 on the left and −3 on the right. .
And we know −9 is less than −3. .
The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

Properties of inequality

Subtraction Property of Inequality Addition Property of Inequality For any numbers a , b , and c , For any numbers a , b , and c , if a < b then a − c < b − c . if a > b then a − c > b − c . if a < b then a + c < b + c . if a > b then a + c > b + c .

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality x + 5 > 9 , the steps would look like this:

x + 5 > 9 Subtract 5 from both sides to isolate x . x + 5 − 5 > 9 − 5 Simplify. x > 4

Any number greater than 4 is a solution to this inequality.

Solve the inequality n − 1 2 ≤ 5 8 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Add 1 2 to both sides of the inequality. .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .
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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

p − 3 4 ≥ 1 6

This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.

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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

r − 1 3 ≤ 7 12

This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.

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Solve inequalities using the division and multiplication properties of inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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