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Step 11: Algebra 1

Linear equations, inequalities, and absolute value

Step 11 · Algebra 1 topics

Solve multi-step linear equations and rearrange formulas, solve inequalities and compound inequalities, and solve equations and inequalities with absolute value.

Solving linear equations

Solving a linear equation means finding the value that makes it true. Simplify each side, gather the variable on one side, then undo the operations. A formula can be solved for one of its letters in the same way.

Example: Fractions and parentheses

Solve 23(x−6)=x−7\frac{2}{3}(x - 6) = x - 7.

  1. Multiply both sides by 3: 2(x − 6) = 3x − 21.
  2. So 2x − 12 = 3x − 21.
  3. Subtract 2x and add 21: x = 9.

Answerx = 9

Example: Decimals

Solve 0.4x + 1.2 = 0.1x + 3.

  1. Multiply every term by 10: 4x + 12 = x + 30.
  2. 3x = 18, so x = 6.

Answerx = 6

Example: Solving a formula

Solve P = 2l + 2w for w.

  1. Subtract 2l from both sides: P − 2l = 2w.
  2. Divide both sides by 2.

Answerw=P−2l2w = \frac{P - 2l}{2}

Solving inequalities

Inequalities are solved like equations, except that multiplying or dividing by a negative number reverses the sign. A compound inequality joins two: "and" means both must hold, "or" means at least one must.

Example: Reversing the sign

Solve 5 − 3x ≥ 17.

  1. Subtract 5: −3x ≥ 12.
  2. Divide by −3 and reverse the sign: x ≤ −4.

Answerx ≤ −4

Example: A compound inequality

Solve −3 < 2x + 1 ≤ 9.

  1. Subtract 1 from all three parts: −4 < 2x ≤ 8.
  2. Divide by 2: −2 < x ≤ 4.

Answer−2 < x ≤ 4

−2 < x ≤ 4: open at −2, filled at 4, and everything between.

Absolute value equations

|x| is the distance of x from 0, so it is never negative. |x| = 5 has two solutions, 5 and −5: both are 5 away from 0.

|x| = 5: the two numbers 5 away from 0.

Example: An absolute value equation

Solve |x − 3| = 7.

  1. x − 3 is 7 or −7.
  2. So x = 10 or x = −4.

Answerx = 10 or x = −4

Example: Get it alone first

Solve 2|x + 1| − 4 = 6.

  1. Add 4: 2|x + 1| = 10.
  2. Divide by 2: |x + 1| = 5.
  3. x + 1 = 5 or x + 1 = −5.

Answerx = 4 or x = −6

Example: An absolute value inequality

Solve |x| < 3.

  1. The numbers less than 3 away from 0.

Answer−3 < x < 3

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Solve 3(x − 4) = 2x + 1.

    Show answer

    x = 13

  2. Solve A=12bhA = \frac{1}{2}bh for h.

    Show answer

    h=2Abh = \frac{2A}{b}

  3. Solve −2x + 5 < 11.

    Show answer

    x > −3

  4. Solve 1 ≤ 3x − 2 < 10.

    Show answer

    1 ≤ x < 4

  5. Solve |x + 2| = 9.

    Show answer

    x = 7 or x = −11

  6. Solve |2x − 1| = 5.

    Show answer

    x = 3 or x = −2

  7. Solve |x − 4| ≤ 2.

    Show answer

    2 ≤ x ≤ 6

What next?