7th Grade Math · Unit 4
Compare an equation with the inequality that uses the same numbers.
Same first steps
An equation and an inequality with the same expression start the same way.
Undo addition or subtraction, then multiply or divide.
One number or many
An equation has one solution. An inequality has a solution set.
Open means the boundary is not included. Closed means it is included.
A negative coefficient
Multiplying or dividing by a negative number does not change an equals sign.
It does reverse an inequality symbol.
The first step is the same. One result is a single number.
Undo the addition on both sides. That step is the same for both.
Equation
2x + 3 = 11
2x + 3 − 3 = 11 − 3
2x = 8
Divide both sides by 2.
x = 4
Inequality
2x + 3 > 11
2x + 3 − 3 > 11 − 3
2x > 8
Divide both sides by 2.
x > 4
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 4. Inequality: x > 4.
Undo the addition on both sides. That step is the same for both.
Equation
3x + 1 = 13
3x + 1 − 1 = 13 − 1
3x = 12
Divide both sides by 3.
x = 4
Inequality
3x + 1 < 13
3x + 1 − 1 < 13 − 1
3x < 12
Divide both sides by 3.
x < 4
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 4. Inequality: x < 4.
Undo the addition on both sides. That step is the same for both.
Equation
4x + 2 = 18
4x + 2 − 2 = 18 − 2
4x = 16
Divide both sides by 4.
x = 4
Inequality
4x + 2 ≤ 18
4x + 2 − 2 ≤ 18 − 2
4x ≤ 16
Divide both sides by 4.
x ≤ 4
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 4. Inequality: x ≤ 4.
Undo the addition on both sides. That step is the same for both.
Equation
5x + 5 = 30
5x + 5 − 5 = 30 − 5
5x = 25
Divide both sides by 5.
x = 5
Inequality
5x + 5 ≥ 30
5x + 5 − 5 ≥ 30 − 5
5x ≥ 25
Divide both sides by 5.
x ≥ 5
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 5. Inequality: x ≥ 5.
Undo the addition on both sides. That step is the same for both.
Equation
2x + 7 = 15
2x + 7 − 7 = 15 − 7
2x = 8
Divide both sides by 2.
x = 4
Inequality
2x + 7 > 15
2x + 7 − 7 > 15 − 7
2x > 8
Divide both sides by 2.
x > 4
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 4. Inequality: x > 4.
Undo the addition on both sides. That step is the same for both.
Equation
3x + 4 = 19
3x + 4 − 4 = 19 − 4
3x = 15
Divide both sides by 3.
x = 5
Inequality
3x + 4 < 19
3x + 4 − 4 < 19 − 4
3x < 15
Divide both sides by 3.
x < 5
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 5. Inequality: x < 5.
Undo the addition on both sides. That step is the same for both.
Equation
6x + 3 = 27
6x + 3 − 3 = 27 − 3
6x = 24
Divide both sides by 6.
x = 4
Inequality
6x + 3 ≤ 27
6x + 3 − 3 ≤ 27 − 3
6x ≤ 24
Divide both sides by 6.
x ≤ 4
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 4. Inequality: x ≤ 4.
Undo the addition on both sides. That step is the same for both.
Equation
4x + 5 = 25
4x + 5 − 5 = 25 − 5
4x = 20
Divide both sides by 4.
x = 5
Inequality
4x + 5 ≥ 25
4x + 5 − 5 ≥ 25 − 5
4x ≥ 20
Divide both sides by 4.
x ≥ 5
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 5. Inequality: x ≥ 5.
Undo the addition on both sides. That step is the same for both.
Equation
2x + 9 = 21
2x + 9 − 9 = 21 − 9
2x = 12
Divide both sides by 2.
x = 6
Inequality
2x + 9 > 21
2x + 9 − 9 > 21 − 9
2x > 12
Divide both sides by 2.
x > 6
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 6. Inequality: x > 6.
Undo the addition on both sides. That step is the same for both.
Equation
3x + 6 = 24
3x + 6 − 6 = 24 − 6
3x = 18
Divide both sides by 3.
x = 6
Inequality
3x + 6 < 24
3x + 6 − 6 < 24 − 6
3x < 18
Divide both sides by 3.
x < 6
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 6. Inequality: x < 6.
Undo the addition on both sides. That step is the same for both.
Equation
5x + 2 = 17
5x + 2 − 2 = 17 − 2
5x = 15
Divide both sides by 5.
x = 3
Inequality
5x + 2 ≥ 17
5x + 2 − 2 ≥ 17 − 2
5x ≥ 15
Divide both sides by 5.
x ≥ 3
The equation has one solution. The inequality has a set of solutions.
Answer: Same steps. Equation: x = 3. Inequality: x ≥ 3.
Compare an open inequality, a closed inequality, and an equation.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
2x + 5 = 13
2x + 5 − 5 = 13 − 5
2x = 8
Divide both sides by 2.
x = 4
Open inequality
2x + 5 > 13
2x + 5 − 5 > 13 − 5
2x > 8
Divide both sides by 2.
x > 4
Closed inequality
2x + 5 ≥ 13
2x + 5 − 5 ≥ 13 − 5
2x ≥ 8
Divide both sides by 2.
x ≥ 4
3 does not fit the inequality.
4 is the equation's one solution. It is included only when the inequality is closed.
6 fits the inequality but not the equation.
Answer: Equation: x = 4. Open: x > 4. Closed: x ≥ 4. Inequality only: 6.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
3x + 2 = 14
3x + 2 − 2 = 14 − 2
3x = 12
Divide both sides by 3.
x = 4
Open inequality
3x + 2 < 14
3x + 2 − 2 < 14 − 2
3x < 12
Divide both sides by 3.
x < 4
Closed inequality
3x + 2 ≤ 14
3x + 2 − 2 ≤ 14 − 2
3x ≤ 12
Divide both sides by 3.
x ≤ 4
2 fits the inequality but not the equation.
4 is the equation's one solution. It is included only when the inequality is closed.
5 does not fit the inequality.
Answer: Equation: x = 4. Open: x < 4. Closed: x ≤ 4. Inequality only: 2.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
4x + 1 = 17
4x + 1 − 1 = 17 − 1
4x = 16
Divide both sides by 4.
x = 4
Open inequality
4x + 1 < 17
4x + 1 − 1 < 17 − 1
4x < 16
Divide both sides by 4.
x < 4
Closed inequality
4x + 1 ≤ 17
4x + 1 − 1 ≤ 17 − 1
4x ≤ 16
Divide both sides by 4.
x ≤ 4
3 fits the inequality but not the equation.
4 is the equation's one solution. It is included only when the inequality is closed.
6 does not fit the inequality.
Answer: Equation: x = 4. Open: x < 4. Closed: x ≤ 4. Inequality only: 3.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
5x + 1 = 21
5x + 1 − 1 = 21 − 1
5x = 20
Divide both sides by 5.
x = 4
Open inequality
5x + 1 > 21
5x + 1 − 1 > 21 − 1
5x > 20
Divide both sides by 5.
x > 4
Closed inequality
5x + 1 ≥ 21
5x + 1 − 1 ≥ 21 − 1
5x ≥ 20
Divide both sides by 5.
x ≥ 4
2 does not fit the inequality.
4 is the equation's one solution. It is included only when the inequality is closed.
6 fits the inequality but not the equation.
Answer: Equation: x = 4. Open: x > 4. Closed: x ≥ 4. Inequality only: 6.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
2x + 7 = 19
2x + 7 − 7 = 19 − 7
2x = 12
Divide both sides by 2.
x = 6
Open inequality
2x + 7 > 19
2x + 7 − 7 > 19 − 7
2x > 12
Divide both sides by 2.
x > 6
Closed inequality
2x + 7 ≥ 19
2x + 7 − 7 ≥ 19 − 7
2x ≥ 12
Divide both sides by 2.
x ≥ 6
4 does not fit the inequality.
6 is the equation's one solution. It is included only when the inequality is closed.
8 fits the inequality but not the equation.
Answer: Equation: x = 6. Open: x > 6. Closed: x ≥ 6. Inequality only: 8.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
3x + 5 = 20
3x + 5 − 5 = 20 − 5
3x = 15
Divide both sides by 3.
x = 5
Open inequality
3x + 5 < 20
3x + 5 − 5 < 20 − 5
3x < 15
Divide both sides by 3.
x < 5
Closed inequality
3x + 5 ≤ 20
3x + 5 − 5 ≤ 20 − 5
3x ≤ 15
Divide both sides by 3.
x ≤ 5
3 fits the inequality but not the equation.
5 is the equation's one solution. It is included only when the inequality is closed.
7 does not fit the inequality.
Answer: Equation: x = 5. Open: x < 5. Closed: x ≤ 5. Inequality only: 3.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
6x + 4 = 28
6x + 4 − 4 = 28 − 4
6x = 24
Divide both sides by 6.
x = 4
Open inequality
6x + 4 > 28
6x + 4 − 4 > 28 − 4
6x > 24
Divide both sides by 6.
x > 4
Closed inequality
6x + 4 ≥ 28
6x + 4 − 4 ≥ 28 − 4
6x ≥ 24
Divide both sides by 6.
x ≥ 4
2 does not fit the inequality.
4 is the equation's one solution. It is included only when the inequality is closed.
5 fits the inequality but not the equation.
Answer: Equation: x = 4. Open: x > 4. Closed: x ≥ 4. Inequality only: 5.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
4x + 6 = 22
4x + 6 − 6 = 22 − 6
4x = 16
Divide both sides by 4.
x = 4
Open inequality
4x + 6 < 22
4x + 6 − 6 < 22 − 6
4x < 16
Divide both sides by 4.
x < 4
Closed inequality
4x + 6 ≤ 22
4x + 6 − 6 ≤ 22 − 6
4x ≤ 16
Divide both sides by 4.
x ≤ 4
2 fits the inequality but not the equation.
4 is the equation's one solution. It is included only when the inequality is closed.
6 does not fit the inequality.
Answer: Equation: x = 4. Open: x < 4. Closed: x ≤ 4. Inequality only: 2.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
2x + 1 = 11
2x + 1 − 1 = 11 − 1
2x = 10
Divide both sides by 2.
x = 5
Open inequality
2x + 1 > 11
2x + 1 − 1 > 11 − 1
2x > 10
Divide both sides by 2.
x > 5
Closed inequality
2x + 1 ≥ 11
2x + 1 − 1 ≥ 11 − 1
2x ≥ 10
Divide both sides by 2.
x ≥ 5
3 does not fit the inequality.
5 is the equation's one solution. It is included only when the inequality is closed.
7 fits the inequality but not the equation.
Answer: Equation: x = 5. Open: x > 5. Closed: x ≥ 5. Inequality only: 7.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
3x + 8 = 23
3x + 8 − 8 = 23 − 8
3x = 15
Divide both sides by 3.
x = 5
Open inequality
3x + 8 < 23
3x + 8 − 8 < 23 − 8
3x < 15
Divide both sides by 3.
x < 5
Closed inequality
3x + 8 ≤ 23
3x + 8 − 8 ≤ 23 − 8
3x ≤ 15
Divide both sides by 3.
x ≤ 5
3 fits the inequality but not the equation.
5 is the equation's one solution. It is included only when the inequality is closed.
8 does not fit the inequality.
Answer: Equation: x = 5. Open: x < 5. Closed: x ≤ 5. Inequality only: 3.
The equation has one solution. Each inequality has many solutions.
Open does not include the boundary number. Closed does.
Equation
5x + 3 = 28
5x + 3 − 3 = 28 − 3
5x = 25
Divide both sides by 5.
x = 5
Open inequality
5x + 3 > 28
5x + 3 − 3 > 28 − 3
5x > 25
Divide both sides by 5.
x > 5
Closed inequality
5x + 3 ≥ 28
5x + 3 − 3 ≥ 28 − 3
5x ≥ 25
Divide both sides by 5.
x ≥ 5
3 does not fit the inequality.
5 is the equation's one solution. It is included only when the inequality is closed.
8 fits the inequality but not the equation.
Answer: Equation: x = 5. Open: x > 5. Closed: x ≥ 5. Inequality only: 8.
A negative coefficient flips the inequality symbol, not the equals sign.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−2x + 6 = 2
−2x + 6 − 6 = 2 − 6
−2x = −4
Divide both sides by −2.
x = 2
Inequality
−2x + 6 > 2
−2x + 6 − 6 > 2 − 6
−2x > −4
Divide both sides by −2. The inequality symbol reverses.
x < 2
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 2. Inequality: x < 2.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−3x + 12 = 3
−3x + 12 − 12 = 3 − 12
−3x = −9
Divide both sides by −3.
x = 3
Inequality
−3x + 12 < 3
−3x + 12 − 12 < 3 − 12
−3x < −9
Divide both sides by −3. The inequality symbol reverses.
x > 3
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 3. Inequality: x > 3.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−4x + 8 = 0
−4x + 8 − 8 = 0 − 8
−4x = −8
Divide both sides by −4.
x = 2
Inequality
−4x + 8 ≥ 0
−4x + 8 − 8 ≥ 0 − 8
−4x ≥ −8
Divide both sides by −4. The inequality symbol reverses.
x ≤ 2
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 2. Inequality: x ≤ 2.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−2x + 10 = 4
−2x + 10 − 10 = 4 − 10
−2x = −6
Divide both sides by −2.
x = 3
Inequality
−2x + 10 > 4
−2x + 10 − 10 > 4 − 10
−2x > −6
Divide both sides by −2. The inequality symbol reverses.
x < 3
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 3. Inequality: x < 3.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−5x + 20 = 5
−5x + 20 − 20 = 5 − 20
−5x = −15
Divide both sides by −5.
x = 3
Inequality
−5x + 20 ≤ 5
−5x + 20 − 20 ≤ 5 − 20
−5x ≤ −15
Divide both sides by −5. The inequality symbol reverses.
x ≥ 3
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 3. Inequality: x ≥ 3.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−3x + 9 = 0
−3x + 9 − 9 = 0 − 9
−3x = −9
Divide both sides by −3.
x = 3
Inequality
−3x + 9 > 0
−3x + 9 − 9 > 0 − 9
−3x > −9
Divide both sides by −3. The inequality symbol reverses.
x < 3
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 3. Inequality: x < 3.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−4x + 16 = 4
−4x + 16 − 16 = 4 − 16
−4x = −12
Divide both sides by −4.
x = 3
Inequality
−4x + 16 < 4
−4x + 16 − 16 < 4 − 16
−4x < −12
Divide both sides by −4. The inequality symbol reverses.
x > 3
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 3. Inequality: x > 3.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−6x + 12 = 0
−6x + 12 − 12 = 0 − 12
−6x = −12
Divide both sides by −6.
x = 2
Inequality
−6x + 12 ≥ 0
−6x + 12 − 12 ≥ 0 − 12
−6x ≥ −12
Divide both sides by −6. The inequality symbol reverses.
x ≤ 2
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 2. Inequality: x ≤ 2.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−12x + 7 = 1
−12x + 7 − 7 = 1 − 7
−12x = −6
Multiply both sides by −2.
x = 12
Inequality
−12x + 7 > 1
−12x + 7 − 7 > 1 − 7
−12x > −6
Divide both sides by −12. The inequality symbol reverses.
x < 12
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 12. Inequality: x < 12.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−32x + 9 = 3
−32x + 9 − 9 = 3 − 9
−32x = −6
Multiply both sides by −23.
x = 4
Inequality
−32x + 9 < 3
−32x + 9 − 9 < 3 − 9
−32x < −6
Divide both sides by −32. The inequality symbol reverses.
x > 4
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 4. Inequality: x > 4.
The equation does not flip a symbol.
The inequality does, because both sides are multiplied or divided by a negative number.
Adding or subtracting does not flip either symbol.
Equation
−23x + 8 = 2
−23x + 8 − 8 = 2 − 8
−23x = −6
Multiply both sides by −32.
x = 9
Inequality
−23x + 8 ≥ 2
−23x + 8 − 8 ≥ 2 − 8
−23x ≥ −6
Divide both sides by −23. The inequality symbol reverses.
x ≤ 9
Answer: The equation does not flip a symbol. The inequality does. Equation: x = 9. Inequality: x ≤ 9.
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