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Solve the inequality 4 b 3 ( 3 b ) > 5 ( b 6 ) + 2 b , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.

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Solve the inequality 9 h 7 ( 2 h ) < 8 ( h + 11 ) + 8 h , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.

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Solve the inequality 1 3 a 1 8 a > 5 24 a + 3 4 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Multiply both sides by the LCD, 24, to clear the fractions. .
Simplify. .
Combine like terms. .
Subtract 5 a from both sides to collect the variables on the left. .
Simplify. .
The statement is false! The inequality is a contradiction.
There is no solution.
Graph the solution on the number line. .
Write the solution in interval notation. There is no solution.
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Solve the inequality 1 4 x 1 12 x > 1 6 x + 7 8 , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”

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Solve the inequality 2 5 z 1 3 z < 1 15 z 3 5 , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”

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Translate to an inequality and solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.

[link] shows some common phrases that indicate inequalities.

> <
is greater than is greater than or equal to is less than is less than or equal to
is more than is at least is smaller than is at most
is larger than is no less than has fewer than is no more than
exceeds is the minimum is lower than is the maximum

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twelve times c is no more than 96.

Solution

Translate. .
Solve—divide both sides by 12. .
Simplify. .
Write in interval notation. .
Graph on the number line. .
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Translate and solve. Then write the solution in interval notation and graph on the number line.

Twenty times y is at most 100

This figure shows the inequality 20y is less than or equal to 100, and then its solution: y is less than or equal to 5. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality y is less than or equal to 5 is graphed on the number line, with an open bracket at y equals 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 5, bracket.

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Translate and solve. Then write the solution in interval notation and graph on the number line.

Nine times z is no less than 135

This figure shows the inequality 9z is greater than or equal to 135, and then its solution: z is greater than or equal to 15. Below this inequality is a number line ranging from 14 to 18 with tick marks for each integer. The inequality z is greater than or equal to 15 is graphed on the number line, with an open bracket at z equals 15, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 15 comma infinity, parenthesis.

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Translate and solve. Then write the solution in interval notation and graph on the number line.

Thirty less than x is at least 45.

Solution

Translate. .
Solve—add 30 to both sides. .
Simplify. .
Write in interval notation. .
Graph on the number line. .
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Translate and solve. Then write the solution in interval notation and graph on the number line.

Nineteen less than p is no less than 47

This figure shows the inequality p minus 19 is greater than or equal to 47, and then its solution: p is greater than or equal to 66. Below this inequality is a number line ranging from 65 to 69 with tick marks for each integer. The inequality p is greater than or equal to 66 is graphed on the number line, with an open bracket at p equals 66, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 66 comma infinity, parenthesis.

Got questions? Get instant answers now!

Translate and solve. Then write the solution in interval notation and graph on the number line.

Four more than a is at most 15.

This figure shows the inequality a plus 4 is less than or equal to 15, and then its solution: a is less than or equal to 11. Below this inequality is a number line ranging from 10 to 14 with tick marks for each integer. The inequality a is less than or equal to 11 is graphed on the number line, with an open bracket at a equals 11, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity 11, bracket.

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Key concepts

  • Subtraction Property of Inequality
    For any numbers a, b, and c,
    if a < b then a c < b c and
    if a > b then a c > b c .
  • Addition Property of Inequality
    For any numbers a, b, and c,
    if a < b then a + c < b + c and
    if a > b then a + c > b + c .
  • Division and Multiplication Properties of Inequalit y
    For any numbers a, b, and c,
    if a < b and c > 0 , then a c < b c and a c > b c .
    if a > b and c > 0 , then a c > b c and a c > b c .
    if a < b and c < 0 , then a c > b c and a c > b c .
    if a > b and c < 0 , then a c < b c and a c < b c .
  • When we divide or multiply an inequality by a:
    • positive number, the inequality stays the same .
    • negative number, the inequality reverses .

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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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