1.06 Compound inequalities

Write a compound inequality to represent the solution set.

Describing the compound inequality in words can help us write the solution to the compound inequality algebraically.

Description: x is less than -1 or x is greater than or equal to 2

Compound inequality: x \lt -1 or x \geq 2

Write the solution to the compound inequality in interval notation.

Since the solution to the inequality occurs in two intervals, we should write the two intervals connected with the union symbol.

\left(-\infty, -1\right) \cup \left[2, \infty\right)

The unfilled circle at -1 indicates that -1 is not in the solution set of the compound inequality, which is stated with the inequality x \lt -1.

The filled circle at 2 indicates that 2 is in the solution set of the compound inequality, which is stated with the inequality x \geq 2.

-2.9 \leq -2.4x + 7.3 \lt 3.7

An "and" compound inequality can be solved simultaneously as long as we apply each operation to all parts of the inequality.

By the division property of inequality, division by a negative number requires reversing the inequality symbols.

We can write the inequality in order from its minimum value to its maximum value by the symmetric property of inequality: 1.5 \lt x \leq 4.25. In set notation, this would be \{x \vert -1.5 \lt x \leq 4.25\}. In interval notation, this would be \left( 1.5 , 4.25 \right].

Finally, we can graph it on the number line with an unfilled endpoint at 1.5 because the inequality is not an "equal to" inequality and a filled endpoint at 4.25 because the inequality is an "equal to" inequality.

We could have also written the compound inequality as two inequalities and solved them separately.

Solve the first inequality:

Solve the second inequality:

When writing the final compound inequality, we can graph the solutions on the number line, then write the compound inequality to match the solution set.

First, plot a filled circle at 4.25 and an unfilled circle at 1.5. Then, determine values of x that make both 4.25 \geq x and x \gt 1.5 true.

Based on the number line, the values of x that satisfy both inequalities are between 1.5, (exclusive) and 4.25, (inclusive). The solution to the inequality is 1.5 \lt x \leq 4.25.

-\dfrac{1}{2}x + 8 \lt 2 or 12x + 9 \leq -15

We can solve the two inequalities separately using properties of inequality and graph the solution on the same number line.

Solve the first inequality:

Solve the second inequality:

The solution to the compound inequality is x \gt 12 or x \leq -2. In set notation, this would be \{x \vert x\leq-2, x \gt 12\}. In interval notation, this would be \left(-\infty, -2\right] \cup \left(12, \infty\right). The solution set is shown on the number line:

3.5x+0.75 \lt 13 or -6.25x-8 \geq 4.5

Solve the first inequality:

Solve the second inequality:

When graphing the solutions to the inequality, we see an overlap in the solution sets since the division property of inequality by a negative number required us to reverse the inequality symbol for the second inequality.

Since the statement for the compound inequality is x\lt3.5 or x\leq-2 and the solution set is a single set of values that are strictly less than 3.5, the solution to the inequality is x \lt 3.5 since its solutions will satisfy either inequality.

We could also have written the answer using interval notation or set notation.

• Interval notation: \left(-\infty, 3.5\right)

• Set notation: \{x \vert x \lt 3.5\} or \{x: x\lt3.5\}

In set notation, we can use a vertical line or a colon before writing the inequality; both notations read "the set of x such that x is less than 3.5."

Write a compound inequality to solve for the possible scores Malia can earn on her 4th test in order to earn a B.

Since Malia's scores must be an average of an 82 to an 87, we can write an "and" compound inequality statement to determine the range of test scores she could earn.

Let x= \text{ the final test score}. Since the minimum score she needs is an 82 and the maximum score she needs is an 87, we can write our expression representing the average of the four test scores in between.

After writing a compound inequality to model a real-world situation, it helps to review the inequality written and confirm that it is reasonable for the situation.

Solve the compound inequality from part (a) and interpret its meaning in context.

Since x can be anywhere from 68 to 88, (inclusive) and we know that x represents Malia's score needed on her final test, the solution means that Malia needs anywhere from a 68 to an 88 on her final test in order to earn a B in social studies.

In interval notation, this solution can be written as \left[68, 88\right].

In set notation, this solution can be written as \{x\vert 68\leq x \leq 88\}.

Determine whether x=75 is a viable solution.

Substituting the given value into the inequality will determine if the solution is a solution to the inequality, then making sense of the solution in context will further our decision.

If x=75, then 68 \leq 75 \leq 88 is a true statement.

x=75 means a score of 75 on the test, and since we know the score can be from a 68 to an 88, we can confirm that x=75 is a viable solution.