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Solving Inequalities Interval Notation, Number Line, Absolute Value, Fractions \u0026 Variables – Algebra
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In mathematics, we often encounter equations with one or two variables. Equations are defined as statements involving variables and an equals sign (=) between the right se (right se) and left se (left se) of the equation. Similarly, we can define inequalities or inequalities as statements containing variables and the inequality signs < , > , ≤ or ≥ between the variables and RHS and LHS . Here are some examples of inequalities: 3x+ 7 ≤ 0 , 4y ≥ 9 , 6y + 8x < 7 ;.
How to solve any Inequality Equations ?
Solving an inequality basically refers to the process of obtaining all possible solutions to an inequality. The set of solutions for any inequality can be found by the following steps: Step 1 – Obtain linear/quadratic/absolute inequalities (derive if not given). Step 2 – Collect all terms that involve the variable on one se (left se) of the inequality and the constant term on the other se (right se). Step 3 – Reduce both ses of the inequality to the simplest form to reduce the inequality to the form, which is H. ax + b ≤ 0 or ax² + bx + c ≥ 0, etc. Step 4 – Solve the inequality obtained in the above steps by diving both ses of the inequality by the coefficients of variables. Step 5 – Write the solution set from the previous steps in the form of interval notation on the number line. Step 6 – Always remember that adding or subtracting the same number to both ses of an inequality will never change the sign of the inequality. However, if we dive or multiply by the same negative number on both ses, the sign of the inequality changes. Even if both ses are multiplied or dived by positive numbers, the sign does not change.
Types Of Inequality Equations !!
You may encounter different types of inequalities, as follows: Linear inequalities in one variable In this type of inequality, we usually have only one variable, then the equation takes the form: ax + b ≥ 0 , ax + b ≤ 0, ax + b < 0 and ax + b > 0; linear inequalities in two variables: let a, b and c be nonzero real numbers, and let x, y be variables. Then this inequality can be represented by two variables: ax+by≥c, ax+by≤c, ax+by
List Of Inequalities Rules !!
In this section, we will mainly discuss the rules that any inequality must obey. There are three inequality rules, described below: 1. The same number can be added to or subtracted from both ses of an equation without changing the sign of the inequality. 2. Both ses of an equation can be multiplied or dived by the same number without changing the sign of the inequality. However, when both ses of the equation are multiplied and dived by a negative number, the sign of the inequality is reversed. 3. Any term of the equation can be brought to the other se by changing the sign without affecting the sign of the inequality. Note: Before delving into this topic, it is important to understand the parentheses used to mark intervals on the number line. Always remember that when finding solutions to inequalities, we mainly use two types of parentheses: 1. Parentheses (also known as paratheses) – represent values without endpoints (a point on the x-axis in an open interval). 2. Square brackets – Indicates the value containing the endpoint (a point on the x-axis in the closed interval) For example: if the interval is given by [2,5) ⇔ This indicates that the value x is all Values in the interval 2 to 5, Including the value (point) ‘2’, excluding the point ‘5’ on the number line, as shown below: Hint: the extreme value on the x-axis, i. H. Periods “infinity” and “-infinity” never append square brackets, i.e. H. The infinity point “∞” should not be connected by square brackets.
Steps to solve linear inequalities In One Variable
In this section, we mainly focus on inequalities in one variable, as shown above. These are mainly in the following forms: ax + b > c or ax + b < c or ax + b ≤ c or ax + b ≥ c; now let's solve some examples by following the steps given above for dealing with inequalities. Suppose we have a linear inequality: 7x + 9 > 30 We have 7x + 9 > 30 ; now, , , 7x > 21 Diving both ses by ‘7’ , we get the next example of inequality: 4x – 16 ≥ 0; Solve for 4x ≥ 16 When ; dive “4” into two ses: that is, x ≥ 4, and draw it on the number line, we finally get the result interval on the number line as [4 , ∞) :.
Steps For Solving Inequalities With Variables On Both Ses !!
In this section, we now conser another variant of linear inequality with variables on both ses of the equation. This is basically of the form: ax + b < cx + d or ax + b > cx + d or ax + b ≥ cx + d or ax + b ≤ cx + d ; now let’s solve the given inequality as follows: 5x – 3 < 3x + 1 Here we first shift all variables on the left and keep the constants on the right. Moving the "3x" term on the left se of the equation, we get: we have 5x – 3x < 4; now, 2x < 4; finally, when both ses are dived by '2': i.e. x < 2, when plotting on the number line When: We end up with the result interval on the number line as (2, ∞) as the solution set of the given inequality with both ses of the variable.
How To Solve Inequalities With Fractions And Variables In The Denominator ?
Whenever we have fractions and rational functions/variables in the denominator, we need to apply some extra steps to solve these inequalities. Fractions refer primarily to equations of the form p/q, i.e.: < k or > > k or ≤ k or ≤ k ; to solve an inequality with fractions and variables in the denominator: 1. First determine the equation (the inequality equation) 2. Now move all terms on the left. and simplify the left se of the equation to: < 0 or >> 0 or ≤ 0 or ≥ ; 3. Make the “x” coefficients in the numerator and denominator positive (if not already done). 4. Now find the critical point by equating the numerator and denominator to “0” respectively. 5. Now enter the critical point on the number line. Finally, dive the number line into different regions based on critical points. 6. Now the rightmost area of the expression on the number line is always positive, and all other areas are alternately positive and negative. So first highlight a plus sign in the rightmost area, then alternately highlight the plus and minus signs in other areas 7. Now we need to select the matching area from the number line based on the inequality sign. Finally, write the range in interval notation. Let’s tackle some examples to further illustrate the above steps. Suppose we want to solve the following inequality: > 2 ; here, there are fractions and variables in the denominator. Just follow the steps listed above: from step 2 above, we move all terms to the left, which is H. – 2 > 0 ; > 0 ; then , , > 0 ; now 0 ; as mentioned in step 3 above, we need positive coefficients for “x”. Therefore, if you multiply both ses of the above inequality by “-1”, the sign of the inequality also changes as follows: Finally, the inequality is: < 0 Now we need to find the critical point of each from step 4 above. In the numerator we have the "x + 12" term. Equating it with '0' gives us the critical point, x + 12 = 0; then, x = -12; similarly we have the term 'x + 5' in the denominator, equating it with '0' we Get another critical point, namely x + 5 = 0; then, x = -5; we get the critical point x = -12, -5. Now we plot these points on the number line. The sol line is dived into three areas as follows: As in step 6 above, we start by marking (+/-) signs for each area on the sol line. So we start with a plus sign in the rightmost region and alternately follow a minus sign and a plus sign: Note: The plus and minus signs above mean that the value of a given inequality equation is "all positive" or "all negative"" The value of that particular interval. Obviously, the set of solutions for a given inequality, i.e.; is the area with a set of all negative values of d. H. The interval (-12,-5) Note: We cannot include these points because they are either There is no equal sign between. Also at x = -5, the specified function becomes undefined (because the denominator becomes zero).
How To Solve Quadratic Inequalities ?
In the case of quadratic equations, we must again follow the concept of critical points (or roots) mentioned above. Here are the steps to follow: 1. First create the quadratic equation (inequalities) 2. Now move all terms on the left. and simplifying the left-hand se of the equation to obtain a result of the form: ax² + bx + c ≥ 0 , ax² + bx + c ≤ 0 , ax² + bx + c < 0 and ax² + bx + c > 0 ; 3. Let “x² ” has a positive coefficient. 4. Now find the critical point by equaling to 0. 5. Now enter the critical point on the number line. Finally, dive the number line into different regions based on critical points. 6. Now the rightmost area of the expression on the number line is always positive, and all other areas are alternately positive and negative. So first highlight a plus sign in the rightmost area, then alternately highlight the plus and minus signs in the other areas 7. Now we need to select the matching area from the number line based on the inequality sign. Finally, write the range in interval notation. Let’s find the solution to the given quadratic inequality: x² + 5x – 15 ≤ -21 ; from step 1, move everything to the left: x² + 5x – 15 + 21 ≤ 0; then we have the quadratic inequality is: x² + 5x + 6 ≤ 0 ; as mentioned in step 3 above, we want a positive coefficient for “x²”. (It’s already positive here, don’t worry) Now, as in step 4, we find the roots of the given quadratic inequality: Using factorization, we can write: (x + 2)(x + 3) ≤ 0; thus , we have two critical points x = -2 , -3 for this quadratic equation. We now draw these points on the number line. As mentioned in step 6 above, we start by marking each region with a flag. So we start with the plus sign for the rightmost region, and alternate the minus and plus signs: now we clearly have the three independent regions above, where the set of solutions needed for a given quadratic inequality are the points -2 and – 3 (gives a negative value). So the solution set is given by [-2,-3]. Note: We include these two points here because there is an equal sign between them in the inequality above.
How To Solve Inequalities With Absolute Values ?
If we have absolute values, we need to use some standard results to solve inequalities including absolute values as shown below. Let a , b , and r be any positive real numbers, then we get the following result, which is true for every absolute value equation: 1. |x| < a ⇔ -a < x < a d. H. x ∈ (-a,a) 2. |x| ≤ a ⇔ -a ≤ x ≤ a d. H. x ∈ [-a,a] 3. |x| > a ⇔ x < -a or x > a 4. |x| ≥ a ⇔ x ≤ -a or x ≥ a 5. |x – a| < r ⇔ a – r < x < a + r d. H. x ∈ (a – r , a + r) 6. |x – a| > r ⇔ x < a – r or x > a + r 7. |x – a| ≤ r ⇔ a – r ≤ x ≤ a + rd. H. x ∈ [a – r , a + r] 8. |x – a| ≥ r ⇔ x ≤ a – r or x ≥ a + r 9. a < |x| < b ⇔ x ∈ (-b , -a ) ∪ (a,b) 10. a ≤ |x| ≤ b ⇔ x ∈ [-b , -a] ∪ [a,b] 11. a ≤ |x + c| ≤ b ⇔ x ∈ [-b + c , -a + c] ∪ [a + c,b + c] 12. a < |x + c| < b ⇔ x ∈ (-b + c , -a + c) ∪ (a + c,b + c) Proof of the above inequalities is beyond the scope of this article (otherwise this article will definitely grow: -p) Now whenever you find any inequality (including absolute value), you just use the above result immediately. Let's solve some inequality examples to make this concept clearer. Let's solve the inequality with the absolute value of: |x – 2| ≥ 5 Here we have the absolute value, which can be solved with result 8 above: i.e. |x – a| ≥ r ⇔ x ≤ a – r or x ≥ a + r; now here we have 'a' = 2 and r = 5, plug these values into the above result to solve the absolute value equation: we get |x – 2| ≥ 5 ⇔ x ≤ 2 – 5 or x ≥ 2 + 5 ; thus |x – 2| ≥ 5 ⇔ x ∈ (-∞,-3] ∪ [7,∞) Finally, the solution set of the inequality given above is , , x ∈ (-∞,-3] ∪ [ 7,∞) , now we solve the example of the absolute value inequality as a mixture of multiple inequalities: Let the given inequality be |x – 1| ≤ 5 , |x| ≥2; the given absolute value inequality system is: |x – 1| ≤ 5 and |x| ≥ 2, now in the inequality: |x – 1| ≤ 5 ⇔ |x – a| ≤ r From result 7 above: |x – a| ≤ r ⇔ a – r ≤ x ≤ a + r d. H. x ∈ [a – r , a + r] where we have a = 1 and r = 5 , substituting the values into the above formula: |x – a| ≤ r ⇔ a – r ≤ x ≤ a + r = 1 – 5 ≤ x ≤ 1 + 5d. H. x ∈ [-4 , 6] We also have a second absolute inequality: |x| ≥ 2; according to result 4 above, |x| ≥ a ⇔ x ≤ -a or x ≥ a ; here we have a = 2 d. H. |x| ≥ a ⇔ x ≤ -a or x ≥ a = x ≤ -2 or x ≥ 2; finally the solution set of the above absolute inequality is: x ∈ (-∞ , -2] ∪ [2 , ∞) The sets of solutions for the two absolute inequalities above are combined. Also for the final solution theorem of the resulting inequalities, we have to combine the two and the common domain will be the final solution theorem as follows: So the solution theorem for the given theorem for a given system of inequalities is [-4, -2] ∪ [2 , 6] Let Now let us solve another inequality example as follows: the given inequality is: > 1 for x ≠ 4; now the given absolute inequality consists of a fractional value which also contains an absolute value. So , , > 1 , , now if we multiply the terms “|x – 4|” on both ses of the equation, we get: d. H. 2 > |x – 4| in the form |x – a| < r ⇔ a – r < x < a + r d. H. x ∈ (a – r , a + r) Using result 5 above, we have: |x – a| < r ⇔ a – r < x < a + r d. H. x ∈ (a – r , a + r) , where a = 4 and r = 2 Substitute the values of 'a' and 'r' into the result above: | x - a | < r ⇔ a – r < x < a + r = 4 – 2 < x < 4 + 2 , i.e. x ∈ (2 , 6) so the solution set for the absolute value inequality example given above is x ∈ (2 , 4 ) ∪ (4 , 6) also x ≠ 4 (since the inequality becomes indeterminate at x = 4) We can also solve arbitrary inequality equations using graphs. In the next article, we will discuss this topic in detail, as well as different ways to quickly draw inequalities. I hope you really enjoyed this article on a sol understanding of inequalities and steps to solve inequalities including fractions. variables, absolute values, etc. If in doubt, let me know in the comments section. report this ad
How do you solve inequalities with fractions and variables in the numerator?
To solve a rational inequality, you first find the zeroes (from the numerator) and the undefined points (from the denominator). You use these zeroes and undefined points to divide the number line into intervals. Then you find the sign of the rational on each interval.
What is the formula of quadratic inequality?
To solve a quadratic inequality, we also apply the same method as illustrated in the procedure below: Write the quadratic inequality in standard form: ax2 + bx + c where a, b and are coefficients and a ≠ 0. Determine the roots of the inequality. Write the solution in inequality notation or interval notation.
How do you simplify absolute value inequalities?
Isolate the absolute value expression on the left side of the inequality. If the number on the other side of the inequality sign is negative, your equation either has no solution or all real numbers as solutions. Use the sign of each side of your inequality to decide which of these cases holds.
How do you solve absolute value equations with variables outside?
Eliminate the absolute value by creating two equations from the expression, representing the positive and negative possibilities for the terms within the bars. Solve for both answers. Practice by solving the absolute value equation 2|x – 4| + 8 = 10 by first subtracting 8 from both sides: 2|x – 4| = 2.
What are the rules for absolute value?
The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
How do you divide similar fractions?
Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators.
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