Write the following in interval notation. x ≥ 0 x\ge0

[ 0 , ∞ ) \left\lbrack0,\infty\right)

( − ∞ , 0 ) \left(-\infty,0\right)

( 0 , ∞ ) \left(0,\infty\right)

( − ∞ , 0 ] \left(-\infty,0\right\rbrack

Write the following in interval notation. x < 7 x<7

( − ∞ , 7 ) \left(-\infty,7\right)

( − ∞ , 7 ] \left(-\infty,7\right\rbrack

[ 7 , ∞ ) \left\lbrack7,\infty\right)

( 7 , ∞ ) \left(7,\infty\right)

What is interval notation and how is it used to express solutions of linear inequalities?

Interval notation is a concise way to express the solution set of a linear inequality as an interval on the number line. It uses parentheses ( ) and brackets [ ] to indicate whether endpoints are excluded or included, respectively. For example, the inequality x > 3 is written as ( 3 , ∞ ) because 3 is not included (parenthesis) and infinity is always excluded. If the inequality is x ≥ 3 , it is written as [ 3 , ∞ ) to show inclusion of 3 (bracket). Interval notation provides a clear and standardized way to communicate solution sets.

2. Linear Equations and Inequalities

Solving Linear Inequalities

2. Linear Equations and Inequalities

Solving Linear Inequalities: Videos & Practice Problems

Learn Concepts Worksheet

Topic summary

Linear inequalities, expressed as $ax + b > c$ or with other inequality symbols, have solutions represented by ranges rather than single values. These solutions can be depicted using set builder notation, interval notation, or number lines with parentheses or brackets indicating inclusion or exclusion of boundary points. Solving inequalities involves isolating the variable using addition, subtraction, multiplication, or division, with the critical rule that multiplying or dividing by a negative number reverses the inequality symbol. Three-part inequalities extend this by applying operations to all three parts, maintaining the variable isolated between two bounds.

concept

Introduction to Linear Inequalities

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Introduction to Linear Inequalities Video Summary

Linear inequalities are closely related to linear equations but differ by using inequality symbols instead of an equal sign. While a linear equation typically takes the form $ax + b = c$, a linear inequality replaces the equal sign with symbols such as $>$, $<$, $\geq$, or $\leq$, resulting in expressions like $ax + b > c$. This subtle change transforms the problem from finding a single solution to identifying a range of solutions.

In a linear equation, solving for $x$ yields a specific value that satisfies the equation. For example, solving \$2x - 6 = 0$ gives $x = 3$. However, when dealing with a linear inequality such as $2x - 6 > 0$, the solution is not a single number but all values of $x$ that make the inequality true. In this case, solving the inequality results in $x > 3\(, meaning any number greater than 3 satisfies the inequality.

To verify solutions of linear inequalities, you can substitute values from the solution range back into the inequality. For instance, plugging in \)x = 5$ into $2x - 6 > 0$ yields $2(5) - 6 = 10 - 6 = 4$, and since $4 > 0$ is true, $x = 5\( is indeed a valid solution. This demonstrates that the solution to a linear inequality is a set of values rather than a single point.

Understanding linear inequalities involves recognizing their standard form \)ax + b \; \square \; c$, where $\square$ represents any inequality symbol. The solutions to these inequalities form intervals or ranges on the number line, contrasting with the single solutions typical of linear equations. This foundational knowledge sets the stage for exploring various methods to solve and graph linear inequalities, highlighting their similarities and differences with linear equations.

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concept

Representing Solutions to Linear Inequalities

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Representing Solutions to Linear Inequalities Video Summary

When solving inequalities, the solution is not a single value but a range of values. Understanding how to represent and visualize these solution sets is essential. One common way to express inequalities is through set builder notation. For example, the inequality "x is greater than 3" can be written as the set of all x such that x > 3, denoted by $\{ x \mid x > 3 \}$. The vertical bar means "such that," and the curly brackets enclose the entire solution set.

Another way to visualize inequalities is on a number line. To represent x > 3, you start at 3 and draw an arrow pointing to the right, indicating all values greater than 3. Since 3 itself is not included, this point is marked with a parenthesis or an open circle to show exclusion.

Interval notation provides a concise way to express these ranges. For x > 3, the interval is written as $(3, \infty)$, where the parenthesis around 3 indicates it is not included, and infinity always uses a parenthesis because it is not a finite value and cannot be included.

When the inequality includes the boundary value, such as x ≥ 3, the notations adjust accordingly. In set builder notation, it becomes $\{ x \mid x \geq 3 \}$. On the number line, the point at 3 is included, represented by a closed circle or a square bracket. In interval notation, this is written as $[3, \infty)$, where the square bracket indicates inclusion of 3.

For inequalities where x is less than a value, such as x < 3, the same principles apply but the arrow on the number line points to the left. The set builder notation is $\{ x \mid x < 3 \}$, and the interval notation is $(-\infty, 3)$ with parentheses indicating exclusion of both endpoints. If the inequality is x ≤ 3, then the set builder notation is $\{ x \mid x \leq 3 \}$, the number line uses a closed circle or square bracket at 3, and the interval notation is $(-\infty, 3]$.

It is important to remember that infinity is never included in interval notation, so it is always enclosed in parentheses. Additionally, on number lines, open circles denote excluded values, while closed circles denote included values. Although parentheses and square brackets are more commonly used in graphs, recognizing open and closed circles is helpful for interpreting inequalities visually.

Mastering these notations—set builder, number line, and interval notation—enables clear communication and understanding of solution sets for linear inequalities, which is foundational for solving and graphing inequalities effectively.

Problem

Which inequality matches the graph?

$x is greater than or equal to 1$

$x is less than or equal to 1$

$x is greater than 1$

$x is less than 1$

Problem

Which inequality matches the graph?

$x\ge1$

$x\le1$

$x>1$

$x<1$

Problem

Use the number line to graph the following inequality.

$x is greater than or equal to 6$

Number line from -8 to 8 with a green arrow starting at 6 (closed circle) extending right, representing x ≥ 6.

Number line from -8 to 8 with a green arrow starting at 6 and extending right, indicating x is greater than or equal to 6.

Number line from -8 to 8 with a closed circle at 6 and a green arrow extending right, representing x ≥ 6.

Number line from -8 to 8 with a closed circle at 6 and a green arrow extending right, representing x greater than or equal to 6.

Problem

Use the number line to graph the following inequality.

$x is greater than or equal to negative 7$

Number line from -8 to 8 with a closed circle at -7 and a green arrow extending right, representing x ≥ -7.

Number line from -8 to 8 with a green arrow starting at -7 and extending right, indicating x is greater than or equal to -7.

Number line from -8 to 8 with a closed circle at -7 and a green arrow extending right, indicating x is greater than or equal to -7.

Number line from -8 to 8 with a green bracket and arrow highlighting all values greater than or equal to -7.

Problem

Write the following in interval notation.
$x is greater than or equal to 0$

$open paren negative infinity comma 0 close paren$

$open bracket 0 comma infinity close paren$

$open paren 0 comma infinity close paren$

$open paren negative infinity comma 0 close bracket$

Problem

Write the following in interval notation.
$x is less than 7$

$open paren negative infinity comma 7 close bracket$

$open bracket 7 comma infinity close paren$

$open paren 7 comma infinity close paren$

$open paren negative infinity comma 7 close paren$

Problem

Rewrite the following as an inequality statement.
$open paren negative infinity comma 2 close bracket$

$x is less than or equal to 2$

$x is less than 2$

$x is greater than or equal to 2$

$x is greater than 2$

Problem

Rewrite the following as an inequality statement.
$open paren 1 comma infinity close paren$

$x is greater than or equal to 1$

$x is less than or equal to 1$

$x is greater than 1$

$x is less than 1$

example

Introduction to Linear Inequalities Example 1

Video duration:

Introduction to Linear Inequalities Example 1 Video Summary

Understanding how to translate between inequalities, interval notation, and graphical representations is essential for mastering the concept of inequalities. When given an inequality such as x > -2, it corresponds to the interval notation (-2, ∞), where the parenthesis indicates that -2 is not included in the set. This matches a graph starting just after -2 and extending infinitely to the right.

For inequalities that include the boundary value, such as x ≥ -5, the interval notation uses a square bracket to denote inclusion: [-5, ∞). This means the set includes -5 and all values greater than -5, which is reflected in the graph by a solid point at -5 and an arrow extending to infinity.

Intervals like (4, ∞) represent values greater than 4 but not including 4 itself, indicated by a parenthesis. The corresponding graph starts just after 4 and extends to the right, without including the point at 4.

When dealing with values less than a certain number, such as values less than -2, the interval notation is (-∞, -2), where the parenthesis again shows that -2 is not included. The graph for this interval starts from negative infinity and ends just before -2, with an open circle at -2.

Intervals that include the endpoint, like (-∞, 3], denote all values less than or equal to 3. The square bracket at 3 indicates inclusion, and the graph reflects this with a solid point at 3 and an arrow extending to the left towards negative infinity.

Finally, inequalities such as -1 ≤ x can be rewritten as x ≥ -1, showing that x is greater than or equal to -1. The interval notation is [-1, ∞), and the graph starts at -1 with a solid point and extends infinitely to the right.

By recognizing the symbols used in interval notation—parentheses ( ) for exclusion and brackets [ ] for inclusion—and understanding how these relate to inequalities and their graphical representations, one can confidently translate between these forms. This skill is fundamental in algebra and helps in solving and visualizing inequality problems effectively.

concept

Solving Linear Inequalities

Video duration:

Solving Linear Inequalities Video Summary

Solving linear inequalities involves isolating the variable using properties similar to those applied in linear equations, but with attention to the inequality symbols. The addition and subtraction properties of inequalities state that if A is greater than B, then adding or subtracting the same value C to both sides preserves the inequality:
$A > B \implies A + C > B + C$ and $A - C > B - C$.

For example, to solve $x - 3 > 11$, add 3 to both sides to isolate x, resulting in $x > 14$. This process mirrors solving linear equations but maintains the inequality symbol.

The multiplication and division properties of inequalities state that if A is greater than B, then multiplying or dividing both sides by a positive number C preserves the inequality:
$A > B \implies A \times C > B \times C$ and $A / C > B / C$ (for $C > 0$).

For instance, solving \$2 < \frac{x}{5}$ involves multiplying both sides by 5, yielding $10 < x\(. The variable can be isolated on either side as long as the inequality remains true.

However, a critical difference arises when multiplying or dividing by a negative number. In such cases, the inequality symbol must be reversed to maintain a true statement. Formally, if A is greater than B and C is negative, then:
\)A > B \implies A \times C < B \times C$ and $A / C < B / C$ (for $C < 0\().

For example, solving \)-7x \geq 21$ requires dividing both sides by $-7$. Since $-7$ is negative, the inequality flips, resulting in $x \leq -3$. This flip accounts for the sign change caused by multiplication or division by a negative number, ensuring the solution remains valid.

In summary, solving linear inequalities involves isolating the variable using addition, subtraction, multiplication, and division properties similar to linear equations. The key distinction is that when multiplying or dividing by a negative number, the inequality symbol must be reversed to preserve the inequality's truth. Mastery of these principles allows for accurate solutions to linear inequalities across various contexts.

example

Solving Linear Inequalities Example 2

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Solving Linear Inequalities Example 2 Video Summary

To solve the linear inequality x + 8 ≤ 12 - x, start by gathering all variable terms on one side and constants on the other. Add x to both sides to eliminate the negative x on the right, resulting in 2x + 8 ≤ 12. Next, subtract 8 from both sides to isolate the variable term: 2x ≤ 4. Finally, divide both sides by 2 to solve for x, giving x ≤ 2.

Graphing this solution on a number line involves marking the point at 2 and shading all values less than or equal to 2. Since the inequality includes equality (≤), use a closed circle or square bracket at 2 to indicate that 2 is part of the solution set.

Expressed in interval notation, the solution is (-∞, 2]. Here, the parenthesis around negative infinity indicates that it is not a specific number and cannot be included, while the square bracket at 2 shows that 2 is included in the interval.

This process demonstrates how to solve linear inequalities by applying inverse operations, similar to solving linear equations, while paying attention to inequality symbols to correctly represent the solution set both graphically and in interval notation.

Problem

Solve the following linear inequalities using the addition and subtraction properties of equality.
$5 is less than or equal to y plus 3$

$2 is greater than y$

$8 is less than or equal to y$

$2 is less than or equal to y$

$2 is greater than or equal to y$

Problem

Solve the following linear inequalities using the addition and subtraction properties of equality.
$x minus 2.3 is less than or equal to 4.1$

$x is greater than or equal to 6.4$

$x is greater than 6.4$

$x is less than or equal to 6.4$

$x is less than or equal to 1.8$

Problem

Solve the following linear inequalities using the addition and subtraction properties of equality.
$7 x + 3 < 2 x + 13$

$x is less than 2$

$x is greater than 2$

$x is less than 1.8$

$x is less than or equal to 1.8$

Problem

Solve the following linear inequalities and write the solution in interval notation.
$2 (x + 4) \leq 3 (x - 1) + x$

$\frac{11}{2} \geq x$

$\frac{11}{2} \leq x$

$\frac{11}{4} \leq x$

$\frac{11}{4} \geq x$

Problem

Solve the following linear inequalities and write the solution in interval notation.
$negative five sixths x is less than 3$

$x < - \frac{5}{2}$

$x < - \frac{18}{5}$

$x > - \frac{18}{5}$

$x > - \frac{5}{2}$

example

Solving Linear Inequalities Example 3

Video duration:

Solving Linear Inequalities Example 3 Video Summary

When solving inequalities, it is important to isolate the variable terms on one side and the constants on the other. Consider the inequality \$2x - 5 < 2x + 10$. By subtracting $2x$ from both sides, the variable terms cancel out, leaving $-5 < 10$. Since this statement is always true, the inequality holds for all real numbers. This means the solution set includes every real number because no matter what value of $x\( is substituted, the inequality remains valid.

In contrast, examine the inequality \)x + 2 \geq x + 8$. Subtracting $x$ from both sides eliminates the variable terms, resulting in $2 \geq 8$. This is a false statement, indicating that the inequality has no solution. Such inequalities are called inconsistent because no value of $x$ can satisfy them.

These examples illustrate key concepts in solving linear inequalities, similar to solving linear equations. An inequality can be an identity, true for all values of the variable, or inconsistent, with no possible solutions. Understanding these outcomes helps in analyzing and interpreting inequalities effectively.

concept

Solving Three Part Inequalities

Video duration:

Solving Three Part Inequalities Video Summary

Solving three-part inequalities involves working with an inequality where the variable expression is situated between two inequality symbols, such as in the example: −14 ≤ 2x − 10 < 2. This type of inequality has three parts: the left side, the center (which contains the variable), and the right side. The key to solving these inequalities is to isolate the variable in the center by performing the same operation on all three parts of the inequality simultaneously.

To solve a three-part inequality, start by eliminating constants or coefficients around the variable just as you would in a simple inequality. For instance, if the variable term is 2x − 10, add 10 to all three parts to cancel out the −10, resulting in −14 + 10 ≤ 2x ≤ 2 + 10, which simplifies to −4 ≤ 2x < 12. Next, divide all three parts by 2 to isolate x, yielding −2 ≤ x < 6. This final inequality shows the variable x bounded between two numbers.

The solution to a three-part inequality can be expressed in set-builder notation as { x | −2 ≤ x < 6 }, which reads as "the set of all x such that x is greater than or equal to −2 and less than 6." In interval notation, this solution is written as [−2, 6), where the square bracket [ indicates that −2 is included in the solution (because of the "less than or equal to" symbol), and the parenthesis ) indicates that 6 is not included (due to the strict "less than" symbol).

Graphically, this solution is represented on a number line by shading the region between −2 and 6. A closed bracket or filled circle is placed at −2 to show inclusion, while an open parenthesis or open circle is placed at 6 to show exclusion. Unlike inequalities involving infinity, there are no arrows extending beyond these points since the solution is bounded.

Understanding three-part inequalities reinforces the concept that any operation performed on one part of the inequality must be applied to all parts to maintain balance. This ensures the variable remains isolated in the center, making it easier to interpret and express the solution. Mastery of this process allows for confident handling of compound inequalities and their representations in various mathematical forms.

Problem

Solve the following inequalities and express the answer in interval notation.
$- 4 \leq 2 x + 1 \leq 7$

$\left\lbrack-6,8\right\rbrack$

$\left\lbrack-\frac32,4\right\rbrack$

$\left\lbrack-\frac52,3\right\rbrack$

$\left\lbrack-10,6\right\rbrack$

Problem

Solve the following inequalities and express the answer in interval notation.
$1 < \frac{x - 2}{2} \leq 4$

$\left\lbrack4,10\right)$

$\left(4,10\right\rbrack$

$\left\lbrack0,6\right)$

$\left(0,6\right\rbrack$

Problem

Solve the following inequalities and graph the solution.
$- 6 < 2 x - 4 < 4$

Number line from -8 to 8 with a red bracket highlighting the interval from -1 to 4.

Number line from -8 to 8 with a red line segment and open circles highlighting the interval between -3 and 1.

Number line from -8 to 8 with a red open interval highlighted between -5 and 0.

Number line from -8 to 8 with the interval between 0 and 5 highlighted in red parentheses.

Problem

Solve the following inequalities and graph the solution.
$- 6 \leq 0.5 (x - 4) < - 3$

Number line from -8 to 8 with a red bracket indicating the interval from -4 inclusive to -1 exclusive.

Number line from -8 to 8 with a red bracket highlighting the interval between 1 and 4, including 1 but excluding 4.

Number line from -8 to 8 with a red bracket highlighting the interval from 2 to 8, including 2 but excluding 8.

Number line graph showing the solution to a compound inequality from -6 to -2, including -6 but excluding -2.

Here’s what students ask on this topic:

Solving a linear inequality is similar to solving a linear equation in that you isolate the variable using addition, subtraction, multiplication, or division. However, the key difference is that the solution to a linear inequality is a range of values rather than a single value. Additionally, when multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality symbol to maintain a true statement. This flipping does not occur when solving linear equations. For example, if you have $−7x \geq 21$ , dividing both sides by $-7$ requires flipping the inequality to get $x \leq - 3$ . This ensures the solution set correctly represents all values that satisfy the inequality.

To represent the solution of a linear inequality on a number line, first identify the boundary value(s) from the inequality. Use an open circle (or parentheses) to indicate that the boundary value is not included (e.g., $x > 3$ ), and a closed circle (or square bracket) to show inclusion (e.g., $x \geq 3$ ). Then, draw an arrow extending from the boundary in the direction of the solution set. For example, for $x > 3$ , place an open circle at 3 and draw an arrow to the right, indicating all values greater than 3. This visual helps understand the range of values that satisfy the inequality.

Interval notation is a concise way to express the solution set of a linear inequality as an interval on the number line. It uses parentheses ( ) and brackets [ ] to indicate whether endpoints are excluded or included, respectively. For example, the inequality $x > 3$ is written as $(3, \infty)$ because 3 is not included (parenthesis) and infinity is always excluded. If the inequality is $x \geq 3$ , it is written as $[3, \infty)$ to show inclusion of 3 (bracket). Interval notation provides a clear and standardized way to communicate solution sets.

A three-part linear inequality has the variable expression between two inequality symbols, such as $- 14 \leq 2x - 10 < 2$ . To solve it, perform the same operation on all three parts to isolate the variable in the middle. For example, add 10 to all three sides to eliminate $- 10$ , then divide all three sides by the coefficient of $x$ . The solution will be an inequality with the variable isolated between two numbers, such as $- 2 \leq x < 6$ . This solution can be expressed in set builder notation, interval notation, or graphed on a number line, showing the range of values that satisfy the inequality.

When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality must be reversed to maintain a true statement. This is because multiplying or dividing by a negative number reverses the order of the values. For example, if $a > b$ and you multiply both sides by $- 1$ , the inequality becomes $- a < - b$ . To keep the inequality valid, you flip the symbol from greater than to less than. Forgetting to flip the inequality symbol leads to incorrect solutions. This rule is unique to inequalities and does not apply to equations.

Solutions to linear inequalities can be expressed in three main notations: set builder notation, interval notation, and graphing on a number line. Set builder notation uses curly brackets and a vertical bar to describe the set, for example, ${x | x > 3}$ , read as "the set of x such that x is greater than 3." Interval notation uses parentheses and brackets to indicate excluded or included endpoints, such as $(3, \infty)$ for x > 3, or $[3, \infty)$ for x ≥ 3. Graphing uses open or closed circles and arrows on a number line to visually represent the solution set. Understanding these notations helps communicate solutions clearly and effectively.

Your Beginning Algebra tutors

Solving Linear Inequalities: Videos & Practice Problems

Introduction to Linear Inequalities

Introduction to Linear Inequalities Video Summary

Representing Solutions to Linear Inequalities

Representing Solutions to Linear Inequalities Video Summary

Which inequality matches the graph?

Which inequality matches the graph?

Use the number line to graph the following inequality.

$x is greater than or equal to 6$

Use the number line to graph the following inequality.

$x is greater than or equal to negative 7$

Write the following in interval notation.
$x is greater than or equal to 0$

Write the following in interval notation.
$x is less than 7$

Rewrite the following as an inequality statement.
$open paren negative infinity comma 2 close bracket$

Rewrite the following as an inequality statement.
$open paren 1 comma infinity close paren$

Introduction to Linear Inequalities Example 1

Introduction to Linear Inequalities Example 1 Video Summary

Solving Linear Inequalities

Solving Linear Inequalities Video Summary

Solving Linear Inequalities Example 2

Solving Linear Inequalities Example 2 Video Summary

Solve the following linear inequalities using the addition and subtraction properties of equality.
$5 is less than or equal to y plus 3$

Solve the following linear inequalities using the addition and subtraction properties of equality.
$x minus 2.3 is less than or equal to 4.1$

Solve the following linear inequalities using the addition and subtraction properties of equality.
$7 x + 3 < 2 x + 13$

Solve the following linear inequalities and write the solution in interval notation.
$2 (x + 4) \leq 3 (x - 1) + x$

Solve the following linear inequalities and write the solution in interval notation.
$negative five sixths x is less than 3$

Solving Linear Inequalities Example 3

Solving Linear Inequalities Example 3 Video Summary

Solving Three Part Inequalities

Solving Three Part Inequalities Video Summary

Solve the following inequalities and express the answer in interval notation.
$- 4 \leq 2 x + 1 \leq 7$

Solve the following inequalities and express the answer in interval notation.
$1 < \frac{x - 2}{2} \leq 4$

Solve the following inequalities and graph the solution.
$- 6 < 2 x - 4 < 4$

Solve the following inequalities and graph the solution.
$- 6 \leq 0.5 (x - 4) < - 3$

Here’s what students ask on this topic:

What is the difference between solving a linear equation and a linear inequality?

How do you represent the solution of a linear inequality on a number line?

What is interval notation and how is it used to express solutions of linear inequalities?

How do you solve a three-part linear inequality?

Why do you have to flip the inequality symbol when multiplying or dividing by a negative number?

What are the different notations used to express solutions of linear inequalities?

Your Beginning Algebra tutors

Solving Linear Inequalities: Videos & Practice Problems

Introduction to Linear Inequalities

Introduction to Linear Inequalities Video Summary

Representing Solutions to Linear Inequalities

Representing Solutions to Linear Inequalities Video Summary

Which inequality matches the graph?

Which inequality matches the graph?

Use the number line to graph the following inequality.x≥6x\ge6

Use the number line to graph the following inequality.x≥−7x\ge-7

Write the following in interval notation.x≥0x\ge0

Write the following in interval notation.x<7x<7

Rewrite the following as an inequality statement.(−∞,2]\left(-\infty,2\right\rbrack

Rewrite the following as an inequality statement.(1,∞)\left(1,\infty\right)

Introduction to Linear Inequalities Example 1

Introduction to Linear Inequalities Example 1 Video Summary

Solving Linear Inequalities

Solving Linear Inequalities Video Summary

Solving Linear Inequalities Example 2

Solving Linear Inequalities Example 2 Video Summary

Solve the following linear inequalities using the addition and subtraction properties of equality.5≤y+35\le y+3

Solve the following linear inequalities using the addition and subtraction properties of equality.x−2.3≤4.1x-2.3\le4.1

Solve the following linear inequalities using the addition and subtraction properties of equality.7x+3<2x+137x+3<2x+13

Solve the following linear inequalities and write the solution in interval notation.2(x+4)≤3(x−1)+x2\left(x+4\right)\le3\left(x-1\right)+x

Solve the following linear inequalities and write the solution in interval notation.−56x<3-\frac56x<3

Solving Linear Inequalities Example 3

Solving Linear Inequalities Example 3 Video Summary

Solving Three Part Inequalities

Solving Three Part Inequalities Video Summary

Solve the following inequalities and express the answer in interval notation.−4≤2x+1≤7-4\le2x+1\le7

Solve the following inequalities and express the answer in interval notation.1<x−22≤41<\frac{x-2}{2}\le4

Solve the following inequalities and graph the solution.−6<2x−4<4-6<2x-4<4

Solve the following inequalities and graph the solution.−6≤0.5(x−4)<−3-6\le0.5\left(x-4\right)<-3

Here’s what students ask on this topic:

What is the difference between solving a linear equation and a linear inequality?

What is the difference between solving a linear equation and a linear inequality?

How do you represent the solution of a linear inequality on a number line?

How do you represent the solution of a linear inequality on a number line?

What is interval notation and how is it used to express solutions of linear inequalities?

What is interval notation and how is it used to express solutions of linear inequalities?

How do you solve a three-part linear inequality?

How do you solve a three-part linear inequality?

Why do you have to flip the inequality symbol when multiplying or dividing by a negative number?

Why do you have to flip the inequality symbol when multiplying or dividing by a negative number?

What are the different notations used to express solutions of linear inequalities?

What are the different notations used to express solutions of linear inequalities?

Your Beginning Algebra tutors

Use the number line to graph the following inequality.

$x is greater than or equal to 6$

Use the number line to graph the following inequality.

$x is greater than or equal to negative 7$

Write the following in interval notation.
$x is greater than or equal to 0$

Write the following in interval notation.
$x is less than 7$

Rewrite the following as an inequality statement.
$open paren negative infinity comma 2 close bracket$

Rewrite the following as an inequality statement.
$open paren 1 comma infinity close paren$

Solve the following linear inequalities using the addition and subtraction properties of equality.
$5 is less than or equal to y plus 3$

Solve the following linear inequalities using the addition and subtraction properties of equality.
$x minus 2.3 is less than or equal to 4.1$

Solve the following linear inequalities using the addition and subtraction properties of equality.
$7 x + 3 < 2 x + 13$

Solve the following linear inequalities and write the solution in interval notation.
$2 (x + 4) \leq 3 (x - 1) + x$

Solve the following linear inequalities and write the solution in interval notation.
$negative five sixths x is less than 3$

Solve the following inequalities and express the answer in interval notation.
$- 4 \leq 2 x + 1 \leq 7$

Solve the following inequalities and express the answer in interval notation.
$1 < \frac{x - 2}{2} \leq 4$

Solve the following inequalities and graph the solution.
$- 6 < 2 x - 4 < 4$

Solve the following inequalities and graph the solution.
$- 6 \leq 0.5 (x - 4) < - 3$