2.7 Solve linear inequalities  (Page 5/8)

Solve the inequality $\frac{1}{3}a-\frac{1}{8}a>\frac{5}{24}a\text{}+\frac{3}{4}$ , 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 $5a$ 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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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.

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.

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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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 $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$ .
  if $a>b$ and $c>0$ , then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$ .
  if $a<b$ and $c<0$ , then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$ .
  if $a>b$ and $c<0$ , then $\frac{a}{c}<\frac{b}{c}$ and $ac<bc$ .
  
• When we divide or multiply an inequality by a:
  • positive number, the inequality stays the same .
  • negative number, the inequality reverses .