Linear Inequalities: Videos & Practice Problems

Understanding interval notation is essential for expressing solution sets compactly, especially for inequalities. Closed intervals use square brackets for endpoints included, while open intervals use parentheses for excluded endpoints. When solving linear inequalities, remember to flip the inequality sign when dividing by a negative number. Graphing these solutions involves using closed circles for included values and open circles for excluded ones. Expressing solutions in interval notation enhances clarity, allowing for effective communication of ranges, such as x is greater than or equal to 3, represented as [3, ∞).

Interval Notation

Interval notation provides a compact way to express solution sets, particularly useful for inequalities. This notation uses parentheses and square brackets to indicate whether endpoints are included or excluded from the set. Understanding how to convert inequalities into interval notation is essential for effectively communicating mathematical solutions.

When dealing with closed intervals, such as \(0 \leq x \leq 5\), the endpoints are included in the set. In interval notation, this is represented as \([0, 5]\). The square brackets indicate that both endpoints, 0 and 5, are part of the solution. When graphing this interval on a number line, closed circles are used at the endpoints to signify their inclusion, and a line connects them to show all values in between are included.

In contrast, open intervals occur when the inequalities do not include the endpoints, such as \(0 < x < 5\). This is expressed in interval notation as \((0, 5)\), using parentheses to denote that 0 and 5 are not included in the set. Graphically, open circles are placed at the endpoints, indicating that these values are excluded, while the line connecting them shows that all values in between are part of the solution.

There are also half-open (or half-closed) intervals, which combine both types of endpoints. For example, the inequality \(0 \leq x < 5\) translates to the interval notation \([0, 5)\). Here, the square bracket indicates that 0 is included, while the parenthesis shows that 5 is not. On a number line, this is represented with a closed circle at 0 and an open circle at 5, illustrating the inclusion of 0 and the exclusion of 5.

Lastly, when dealing with inequalities that extend indefinitely, such as \(x \geq 3\), the interval notation is \([3, \infty)\). The square bracket at 3 indicates inclusion, while the parenthesis at infinity signifies that infinity is not a specific endpoint that can be reached. Graphically, this is represented with a closed circle at 3 and an arrow extending to the right, indicating that all values greater than or equal to 3 are included in the solution.

Understanding these concepts of interval notation is crucial for accurately expressing and graphing solution sets in mathematics.

Express the given set in interval notation and graph.
{$x$ | 14 ≤ $x$ < 26}

• A. (14,26)
• B. [14,26]
• C. [14, 26)
• D. (14,26] 

Answer: (C)

Express the given interval in set builder notation and graph. (−∞, 0]

• A. {$x\left|x\right.$ ≤ 0}
• B. {$x\left|x\right.$ < 0}
• C. {$x\left|x\right.$ > 0}
• D. $\left\lbrace x|x\ge0\right\rbrace$

Answer: (A)

Express the given set in interval notation and graph.
$\left\lbrace\right.$$x$ $|$$x\le7$$\}$

• A. (−∞,7]
• B. [−∞,7]
• C. (−∞,7)
• D. [7, ∞)

Answer: (A)

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols such as less than (<) or greater than (>). For example, the inequality 2x - 6 ≤ 0 indicates that we are looking for values of x that make the statement true, rather than just a single solution.

To solve a linear inequality, you can follow the same steps as you would for a linear equation. For instance, starting with 2x - 6 ≤ 0, you can add 6 to both sides to isolate the term with x, resulting in 2x ≤ 6. Dividing both sides by 2 gives you x ≤ 3. This means that any value of x that is less than or equal to 3 satisfies the inequality.

However, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you start with -2x - 6 ≤ 0 and move the 6 to the other side, you get -2x ≤ 6. Dividing both sides by -2 requires flipping the inequality sign, resulting in x ≥ -3. This indicates that any value of x greater than or equal to -3 is a solution.

When graphing the solutions to linear inequalities, the representation differs from that of linear equations. For x ≤ 3, you would place a closed circle at 3 on the number line and shade to the left, indicating all values less than or equal to 3. In contrast, for x ≥ -3, you would place a closed circle at -3 and shade to the right, indicating all values greater than or equal to -3.

In interval notation, the solution x ≤ 3 is expressed as (-∞, 3], where the square bracket indicates that 3 is included in the solution. Similarly, x ≥ -3 is expressed as [-3, ∞), with the square bracket indicating that -3 is included. Remember that infinity is always represented with parentheses.

Understanding how to solve and graph linear inequalities is essential, as it allows you to express a range of solutions rather than just a single value. Practice with various inequalities will help solidify these concepts.

Solve the inequality. Express the solution set in interval notation and graph. $2x+12>19$

• A. $\left(-\infty,\frac72\right)$
• B. $\left(-\infty,\frac72\right]$
• C. $\left[\frac72,\infty\right)$
• D. $\left(\frac72,\infty\right)$

Answer: (D)

Linear Inequalities with Fractions & Variables on Both Sides

When solving linear inequalities that involve fractions or variables on both sides, the approach mirrors that of linear equations. The first step is to eliminate fractions by multiplying the entire inequality by the least common denominator (LCD). For example, consider the inequality:

\( \frac{1}{4}x + 2 \geq \frac{1}{12} - \frac{1}{3}x \)

In this case, the denominators are 4, 12, and 3, and the least common denominator is 12. Multiplying each term by 12 gives:

\( 12 \cdot \frac{1}{4}x + 12 \cdot 2 \geq 12 \cdot \frac{1}{12} - 12 \cdot \frac{1}{3}x \)

This simplifies to:

\( 3x + 24 \geq -1 - 4x \)

Next, distribute and combine like terms. Distributing the 3 results in:

\( 3x + 24 \geq -1 - 4x \)

To isolate the variable, move all terms involving \(x\) to one side and constant terms to the other. Adding \(4x\) to both sides and subtracting 24 from both sides yields:

\( 3x + 4x \geq -1 - 24 \)

This simplifies to:

\( 7x \geq -25 \)

Dividing both sides by 7 gives:

\( x \geq -\frac{25}{7} \)

To express the solution in interval notation, since the endpoint is included, we use a square bracket. The interval notation for this solution is:

\( \left[-\frac{25}{7}, \infty\right) \)

Graphically, this is represented with a solid circle at \(-\frac{25}{7}\) and an arrow extending to the right, indicating all values greater than or equal to \(-\frac{25}{7}\).

Solve the inequality. Express the solution set in interval notation and graph. 
$\frac13\left(x+1\right)\ge\frac15\left(3+2x\right)$

• A. $\left(-4,\infty\right)$
• B. $\left[4,\infty\right)$
• C. $\left[-4,\infty\right)$
• D. $\left(-\infty,-4]\right.$

Answer: (D)

Here’s what students ask on this topic:

What is interval notation and how is it used to express solution sets?

Interval notation is a method of writing subsets of the real number line. It uses parentheses and square brackets to describe intervals. Parentheses, ( and ), indicate that an endpoint is not included in the interval, while square brackets, [ and ], indicate that an endpoint is included. For example, the interval [3, ∞) represents all numbers greater than or equal to 3. This notation is particularly useful for expressing solution sets of inequalities compactly. For instance, the solution to x ≥ 3 is written as [3, ∞) in interval notation.

How do you solve linear inequalities and graph their solutions?

To solve linear inequalities, follow the same steps as solving linear equations: isolate the variable on one side. For example, to solve 2x - 6 ≤ 0, add 6 to both sides to get 2x ≤ 6, then divide by 2 to get x ≤ 3. When graphing, use a closed circle for ≤ or ≥ and an open circle for < or >. Draw a line to the left for ≤ or < and to the right for ≥ or >. For x ≤ 3, place a closed circle at 3 and shade to the left. In interval notation, this is written as (-∞, 3].

What happens when you multiply or divide both sides of an inequality by a negative number?

When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have -2x ≤ 6, dividing both sides by -2 gives x ≥ -3. This rule ensures the inequality remains true. Always remember to reverse the inequality sign to maintain the correct relationship between the values.

How do you express the solution to a linear inequality in interval notation?

To express the solution to a linear inequality in interval notation, identify the range of values that satisfy the inequality. For example, if the solution is x ≥ -1, the interval notation is [-1, ∞). Use square brackets for inclusive endpoints (≤ or ≥) and parentheses for exclusive endpoints (< or >). For x < 4, the interval notation is (-∞, 4). This notation provides a concise way to represent the solution set.

How do you solve linear inequalities with fractions and variables on both sides?

To solve linear inequalities with fractions and variables on both sides, first eliminate the fractions by multiplying the entire inequality by the least common denominator (LCD). For example, for (1/4)x + 2 ≥ (1/12) - (1/3)x, multiply by 12 (the LCD) to get 3x + 24 ≥ 1 - 4x. Then, combine like terms and isolate the variable. Add 4x to both sides to get 7x + 24 ≥ 1, then subtract 24 from both sides to get 7x ≥ -23. Finally, divide by 7 to get x ≥ -23/7. Graph the solution and express it in interval notation as [-23/7, ∞).