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2.7 Solve linear inequalities  (Page 4/8)

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

−24 < 4 3 n

This figure shows the inequality n is greater than negative 18. Below this inequality is a number line ranging from negative 20 to negative 16 with tick marks for each integer. The inequality n is greater than negative 18 is graphed on the number line, with an open parenthesis at n equals negative 18, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 18 comma infinity, parenthesis.

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Solve the inequality t −2 ≥ 8 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Multiply both sides of the inequality by −2 .
Since −2 < 0 , the inequality reverses.		 .
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.

k −12 ≤ 15

This figure shows the inequality k is greater than or equal to negative 180. Below this inequality is a number line ranging from negative 181 to negative 177 with tick marks for each integer. The inequality k is greater than or equal to negative 180 is graphed on the number line, with an open bracket at n equals negative 180, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, negative 180 comma infinity, parenthesis.

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

u −4 ≥ −16

This figure shows the inequality u is less than or equal to 64. Below this inequality is a number line ranging from 62 to 66 with tick marks for each integer. The inequality u is less than or equal to 64 is graphed on the number line, with an open bracket at u equals 64, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 64, bracket.

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Solve inequalities that require simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

Solve the inequality 4 m ≤ 9 m + 17 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Subtract 9 m from both sides to collect the variables on the left. .
Simplify. .
Divide both sides of the inequality by −5, and reverse the inequality. .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .
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Solve the inequality 3 q   ≥   7 q   −   23 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality q is less than or equal to 23/4. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality q is less than or equal to 23/4 is graphed on the number line, with an open bracket at q equals 23/4 (written in), and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 23/4, bracket.

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Solve the inequality 6 x < 10 x + 19 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality x is greater than negative 19/4. Below this inequality is a number line ranging from negative 7 to negative 3, with tick marks for each integer. The inequality x is greater than negative 19/4 is graphed on the number line, with an open parenthesis at x equals negative 19/4 (written in), and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 19/4 comma infinity, parenthesis.

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Solve the inequality 8 p + 3 ( p − 12 ) > 7 p − 28 , graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible. 8 p + 3 ( p − 12 ) > 7 p − 28
Distribute. 8 p + 3 p − 36 > 7 p − 28
Combine like terms. 11 p − 36 > 7 p − 28
Subtract 7 p from both sides to collect the variables on the left. 11 p − 36 − 7 p > 7 p − 28 − 7 p
Simplify. 4 p − 36 > −28
Add 36 to both sides to collect the constants on the right. 4 p − 36 + 36 > −28 + 36
Simplify. 4 p > 8
Divide both sides of the inequality by 4; the inequality stays the same. 4 p 4 > 8 4
Simplify. p > 2
Graph the solution on the number line. .
Write the solution in interal notation. ( 2 , ∞ )
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Solve the inequality 9 y + 2 ( y + 6 ) > 5 y − 24 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality y is greater than negative 6. Below this inequality is a number line ranging from negative 7 to negative 3 with tick marks for each integer. The inequality y is greater than negative 6 is graphed on the number line, with an open parenthesis at y equals negative 6, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 6 comma infinity, parenthesis.

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Solve the inequality 6 u + 8 ( u − 1 ) > 10 u + 32 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality u is greater than 10. Below this inequality is a number line ranging from 9 to 13 with tick marks for each integer. The inequality u is greater than 10 is graphed on the number line, with an open parenthesis at u equals 10, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 10 comma infinity, parenthesis.

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Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

Solve the inequality 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x , graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible. 8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x
Distribute. 8 x − 10 + 2 x < 4 x + 36 + 6 x
Combine like terms. 10 x − 10 < 10 x + 36
Subtract 10 x from both sides to collect the variables on the left. 10 x − 10 − 10 x < 10 x + 36 − 10 x
Simplify. −10 < 36
The x ’s are gone, and we have a true statement. The inequality is an identity.
The solution is all real numbers.
Graph the solution on the number line. .
Write the solution in interval notation. ( −∞ , ∞ )
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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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