Math 1 · Lesson 6 · part 5 of 6
Three worked examples, every line named
The number you add to both sides or multiply both sides by is shown in this color .
Three worked examples, every line named
The number you add to both sides or multiply both sides by is shown in this color. Each example ends with a number line of the answer.
3x - 4 < 11
The original3x - 4 < 11Read it: 3x minus 4 is less than 11. The gator faces the 11, so the 11 is the bigger side. There is a subtraction to rewrite.
Step 1 rewrite3x + (−4) < 11Subtracting 4 is adding −4, its additive inverse (the opposite). Lesson 3 called this translating. Nothing else to do: no written ÷ sign, no lonely x.
Step 23x + (−4) < 11No parentheses, no like terms (the terms with the same letter part) to combine. Step 2 has no work here.
Step 3 opposite of −43x + (−4) + 4 < 11 + 4, so 3x + 0 < 15, that is 3x < 15Add 4, the additive inverse (the opposite) of −4, to both sides. That is the addition property of inequality (add the same thing to both sides, and the sign stays the same). On the left, (−4) + 4 = 0 and 3x + 0 = 3x, by the additive identity (the do-nothing number 0).
Step 3 reciprocal13 × 3x < 13 × 15, so (13 × 3)x < 5, so 1x < 5Multiply both sides by 13, the multiplicative inverse (the reciprocal) of the coefficient (the number on the variable) 3. The number is positive, so the sign stays: that is the multiplication property of inequality for a positive number. The associative property of multiplication (the factors change partners, the grouping moves) regroups it, and 13 × 3 = 1.
Answerx < 51x = x by the multiplicative identity (the do-nothing number 1). Every number smaller than 5 works. The 5 itself does not, because the sign is strict (no equals bar under it).
CheckTry x = 0: 3 × 0 + (−4) = −4, and −4 < 11 is true. Try x = 5, the boundary: 3 × 5 + (−4) = 11, and 11 < 11 is false, so 5 is not included. Try x = 6: 18 + (−4) = 14, and 14 < 11 is false.An inequality has many answers, so check three numbers: one inside, the boundary, and one outside. All three behaved the way x < 5 says they should.
7 - 2x ≥ 1
The original7 - 2x ≥ 1Read it: 7 minus 2x is greater than or equal to 1. The little bar under the sign means the boundary number counts too. There is a subtraction to rewrite.
Step 1 rewrite7 + (−2)x ≥ 1Subtracting 2x is adding −2x, its additive inverse (the opposite). Written properly, that term is (−2)x. The coefficient (the number on the variable) is −2. Keep an eye on it: a negative coefficient is where the new rule will show up.
Step 3 opposite of 77 + (−2)x + (−7) ≥ 1 + (−7), so (−2)x ≥ −6Add −7, the additive inverse (the opposite) of 7, to both sides: the addition property of inequality (add the same thing to both sides, and the sign stays the same). The commutative property of addition (the terms trade places) puts 7 and −7 together, they make 0, and the additive identity (the do-nothing number 0) leaves (−2)x. Adding a negative number does NOT flip the sign. Only multiplying does.
Step 3 reciprocal, and the flip(−12) × (−2)x ≤ (−12) × (−6), so 1x ≤ 3The coefficient (the number on the variable) is −2, so multiply both sides by −12, its multiplicative inverse (the reciprocal). That number is negative, so the sign turns around: ≥ becomes ≤. This is the multiplication property of inequality for a negative number. On the left, (−12) × (−2) = 1. On the right, (−12) × (−6) = 3.
Answerx ≤ 31x = x by the multiplicative identity (the do-nothing number 1). Every number up to and including 3 works.
CheckTry x = 0: 7 + (−2) × 0 = 7, and 7 ≥ 1 is true. Try x = 3, the boundary: 7 + (−2) × 3 = 1, and 1 ≥ 1 is true, so 3 is included. Try x = 5: 7 + (−2) × 5 = −3, and −3 ≥ 1 is false.If you had forgotten to flip, you would have written x ≥ 3, and then x = 5 would have to work. It does not. The check catches a missing flip every time.
4(x + 1) > 6x - 2
The original4(x + 1) > 6x - 2Parentheses on the left, an x term on both sides, and a subtraction on the right.
Step 1 rewrite4(1x + 1) > 6x + (−2)Subtracting 2 is adding −2, its additive inverse (the opposite). The x inside the parentheses is really 1x, by the multiplicative identity (the do-nothing number 1).
Step 2 distribute4x + 4 > 6x + (−2)The distributive property (hand one out to everyone) hands the 4 to the 1x and to the 1: 4 × 1x = 4x and 4 × 1 = 4. No like terms (the terms with the same letter part) to combine on either side.
Step 3 move 4x4x + 4 + (−4x) > 6x + (−2) + (−4x), so 4 > 2x + (−2)The coefficients (the numbers on the variables) are 4 and 6. Move the smaller one, 4x, by adding its additive inverse (the opposite), −4x, to both sides. Moving the smaller x term keeps the coefficient you are left with positive, so you will not need to flip. On the left, 4x + (−4x) = 0x = 0 by the zero rule (0 times anything is 0). On the right, 6x + (−4x) = (6 + (−4))x = 2x, by the distributive property run backwards (factoring, pulling out what the terms share).
Step 3 opposite of −24 + 2 > 2x + (−2) + 2, so 6 > 2xAdd 2, the additive inverse (the opposite) of −2, to both sides: the addition property of inequality (add the same thing to both sides, and the sign stays the same).
Step 3 reciprocal12 × 6 > 12 × 2x, so 3 > 1xMultiply both sides by 12, the multiplicative inverse (the reciprocal) of 2. It is positive, so the sign stays: the multiplication property of inequality for a positive number.
Answer3 > x, which is the same as x < 3Read the gator: it is facing the 3, so 3 is the bigger side and x is the smaller side. Turning the line around to put x first is not a flip. It is the same sentence read from the other end. The sign still eats the 3.
CheckTry x = 0: 4 × (0 + 1) = 4, and 6 × 0 + (−2) = −2. Is 4 > −2? Yes. Try x = 3, the boundary: 4 × 4 = 16, and 18 + (−2) = 16. Is 16 > 16? No, so 3 is not included. Try x = 4: 20 and 22. Is 20 > 22? No.Inside works, the boundary does not, outside does not. That is exactly what x < 3 promises.