Equations and inequalities (Math 1)

High School Math 1 · Unit 4 · multi-step-equations-literals · Teacher edition

multi-step-equations-literals

Name ______________________________Date ______________

1.
Solve $5x - 7 = 3x + 9$ .
Answer: ______________
2.
The area of a triangle is $A = \dfrac{1}{2}bh$ . Solve for $h$ . Enter your answer as an equation beginning with $h =$ .
Answer: ______________
3.
How many solutions does $3(2x - 1) + 4 = 6x + 1$ have?
1. (A)
  no solution
2. (B)
  exactly one solution, $x = 0$
3. (C)
  exactly one solution, $x = 1$
4. (D)
  infinitely many solutions
4.
Which describes all solutions of $-3 < 2x + 1 \le 7$ ?
1. (A)
  $x < -2$ or $x \ge 3$
2. (B)
  $-2 < x \le 3$
3. (C)
  $-1 < x \le 4$
4. (D)
  $-2 \le x < 3$
5.
The equation $a(x - 2) = 3x + b$ has infinitely many solutions. What is $a + b$ ?
1. (A)
  $-9$
2. (B)
  $-6$
3. (C)
  $-3$
4. (D)
  $3$
5. (E)
  $9$

Answer key — Equations and inequalities (Math 1)

1.
8multi-step-equations-literals-01
$5x - 7 = 3x + 9 \Rightarrow 2x = 16 \Rightarrow x = \mathbf{8}$ . Check: $40 - 7 = 33 = 24 + 9$ . ✓
2.
$h=\frac{2A}{b}$ multi-step-equations-literals-02
$A = \frac{1}{2}bh \Rightarrow 2A = bh \Rightarrow \mathbf{h = \dfrac{2A}{b}}$ . Check with $b = 4, h = 3$ : $A = 6$ and $\frac{2 \cdot 6}{4} = 3$ . ✓
3.
(D)
infinitely many solutions
multi-step-equations-literals-03
Left side simplifies to $6x + 1$ , which is identical to the right side. Subtracting $6x$ leaves $1 = 1$ , true for every $x$ : an identity with infinitely many solutions. (“No solution” would be a false statement like $1 = 4$ ; a single value appears only when the $x$ terms do not cancel.)
4.
(B)
$-2 < x \le 3$
multi-step-equations-literals-04
$-3 < 2x + 1 \le 7 \Rightarrow -4 < 2x \le 6 \Rightarrow \mathbf{-2 < x \le 3}$ , a segment open at $-2$ and closed at $3$ . (“ $x < -2$ or $x \ge 3$ ” is two rays — an OR, the wrong connective; $-1 < x \le 4$ adds $1$ instead of subtracting; $-2 \le x < 3$ swaps which end is closed.)
5.
(C)
$-3$
multi-step-equations-literals-05
$ax - 2a = 3x + b$ is an identity when $a = 3$ and $b = -2a = -6$ . So $a + b = 3 + (-6) = \mathbf{-3}$ . ( $9$ takes $b = +6$ ; $-6$ reports $b$ alone; $3$ reports $a$ alone.)